Title: Independence

Q: Created by: Dr. Pang Chung Yin Amy, Dr. Chiang Shun Ling, and Naw Sai.
Department of Mathematics,
Hong Kong Baptist University (HKBU) © 2025.
A: These decks may be freely shared and distributed among students and educators. We hope they help in your linear algebra journey!

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 6 \\ 4 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 6 \\ 4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 3 & 6 & 0 \\ 1 & 3 & 4 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 3 & 2 & 0 \\ 2 & 2 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 8 \\ 4 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 8 \\ 4 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 8 & 2 & 3 & 0 \\ 4 & 2 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0.50 & 0 \\ 0 & 1 & -0.50 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 6 \\ 8 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 6 \\ 8 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 2 & 6 & 0 \\ 2 & 3 & 8 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & 2 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 4 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 4 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 4 & 2 & 2 & 0 \\ 2 & 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0.50 & 0.50 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{2}$ is a free variable, so $c_{2}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 1 & 3 & 0 \\ 2 & 2 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{2}$ is a free variable, so $c_{2}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 2 & 2 & 0 \\ 3 & 0 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 3 & 2 & 0 \\ 2 & 3 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & -1 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 5 \\ 5 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 5 \\ 5 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 5 & 1 & 0 \\ 1 & 5 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & -2 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 0 & 3 & 0 \\ 0 & 1 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & 2 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 5 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 5 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 0 & 3 & 0 \\ 5 & 2 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & -2 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 3 & 3 & 0 \\ 1 & 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 0 & 2 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 2 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 3 & 2 & 0 \\ 0 & 0 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{2}$ is a free variable, so $c_{2}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 6 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 6 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 1 & 1 & 0 \\ 2 & 6 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & -2 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 1 & 1 & 0 \\ 1 & 2 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 1 & 3 & 0 \\ 3 & 3 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{2}$ is a free variable, so $c_{2}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 2 & 1 & 0 \\ 2 & 0 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0.50 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 2 & 2 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$.
Since the number of vectors (3) is greater than the dimension (2), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 6 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 6 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 3 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 0 & 1 & 0 \\ 6 & 3 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0.50 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{2}, c_{3}$ are free variables, so $c_{2}, c_{3}$ can take nonzero values, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 6 \\ 7 \\ 5 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 3 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 2 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 6 \\ 7 \\ 5 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 2 & 6 & 0 \\ 3 & 2 & 7 & 0 \\ 3 & 1 & 5 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 4 \\ 4 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 0 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 4 \\ 4 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 2 & 1 & 0 \\ 0 & 4 & 2 & 0 \\ 3 & 4 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0.50 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 5 \\ 6 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 5 \\ 6 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 5 & 3 & 1 & 0 \\ 6 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0.50 & 0 \\ 0 & 1 & -0.50 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 2 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 3 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 2 & 0 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 6 \\ 3 \\ 9 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 3 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 0 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 6 \\ 3 \\ 9 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 2 & 6 & 0 \\ 3 & 0 & 3 & 0 \\ 3 & 3 & 9 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 6 \\ 5 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 6 \\ 5 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 3 & 1 & 0 \\ 3 & 6 & 0 & 0 \\ 1 & 5 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & -2 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 3 & 2 & 0 \\ 0 & 3 & 3 & 0 \\ 1 & 1 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & -1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 2 & 2 & 0 \\ 0 & 1 & 1 & 0 \\ 3 & 3 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 4 \\ 2 \\ 4 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 4 \\ 2 \\ 4 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 4 & 2 & 0 \\ 2 & 2 & 1 & 0 \\ 1 & 4 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0.50 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 4 \\ 6 \\ 4 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 4 \\ 6 \\ 4 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 4 & 2 & 0 \\ 3 & 6 & 3 & 0 \\ 2 & 4 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0.50 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 0 & 0 & 0 \\ 2 & 1 & 3 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 3 & 2 & 0 \\ 1 & 1 & 2 & 0 \\ 2 & 2 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{2}, c_{3}$ are free variables, so $c_{2}, c_{3}$ can take nonzero values, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 2 & 2 & 0 \\ 2 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 3 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 0 & 0 & 0 \\ 1 & 3 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 3 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{2}, c_{3}$ are free variables, so $c_{2}, c_{3}$ can take nonzero values, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 10 \\ 8 \\ 8 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 10 \\ 8 \\ 8 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 3 & 10 & 0 \\ 1 & 3 & 8 & 0 \\ 2 & 2 & 8 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 2 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 2 & 2 & 0 \\ 3 & 0 & 3 & 0 \\ 1 & 0 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 4 \\ 6 \\ 6 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 4 \\ 6 \\ 6 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 4 & 1 & 0 \\ 2 & 6 & 2 & 0 \\ 2 & 6 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & -0.50 & 0 \\ 0 & 1 & 0.50 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 0 & 2 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 2 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 3 & 2 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
$c_{2}, c_{3}$ are free variables, so $c_{2}, c_{3}$ can take nonzero values, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 5 \\ 7 \\ 8 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 3 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 5 \\ 7 \\ 8 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix} + c_{4}\begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 1 & 5 & 3 & 2 & 0 \\ 3 & 7 & 1 & 3 & 0 \\ 3 & 8 & 2 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & -2 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 5 \\ 10 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 5 \\ 10 \\ 2 \end{bmatrix} + c_{4}\begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 0 & 3 & 5 & 1 & 0 \\ 3 & 1 & 10 & 3 & 0 \\ 1 & 1 & 2 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & -0.50 & 0 \\ 0 & 1 & 0 & -0.50 & 0 \\ 0 & 0 & 1 & 0.50 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 4 \\ 7 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 4 \\ 7 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} + c_{4}\begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 0 & 2 & 2 & 3 & 0 \\ 1 & 4 & 2 & 3 & 0 \\ 2 & 7 & 3 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & -2 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 10 \\ 3 \\ 11 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 10 \\ 3 \\ 11 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 1 \\ 3 \end{bmatrix} + c_{4}\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 2 & 10 & 3 & 1 & 0 \\ 1 & 3 & 1 & 0 & 0 \\ 3 & 11 & 3 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & -0.50 & 0 \\ 0 & 1 & 0 & 0.50 & 0 \\ 0 & 0 & 1 & -1 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 0 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix} + c_{4}\begin{bmatrix} 1 \\ 3 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 2 & 1 & 1 & 1 & 0 \\ 0 & 3 & 3 & 3 & 0 \\ 3 & 2 & 2 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 12 \\ 6 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} + c_{4}\begin{bmatrix} 12 \\ 6 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 3 & 3 & 1 & 12 & 0 \\ 0 & 3 & 0 & 6 & 0 \\ 1 & 0 & 0 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 12 \\ 6 \\ 16 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix} + c_{4}\begin{bmatrix} 12 \\ 6 \\ 16 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 2 & 3 & 1 & 12 & 0 \\ 1 & 1 & 1 & 6 & 0 \\ 3 & 3 & 2 & 16 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 1 & 2 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 5 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 5 \\ 2 \end{bmatrix} + c_{4}\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 0 & 2 & 2 & 3 & 0 \\ 2 & 1 & 5 & 2 & 0 \\ 0 & 2 & 2 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 2 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 8 \\ 12 \\ 7 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 8 \\ 12 \\ 7 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + c_{4}\begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 3 & 8 & 1 & 0 & 0 \\ 3 & 12 & 2 & 2 & 0 \\ 0 & 7 & 3 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & -2 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & -2 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix} + c_{4}\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 3 & 3 & 1 & 1 & 0 \\ 1 & 1 & 3 & 3 & 0 \\ 2 & 2 & 2 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{2}$ is a free variable, so $c_{2}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 6 \\ 12 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 3 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 2 \\ 0 \end{bmatrix} + c_{4}\begin{bmatrix} 0 \\ 6 \\ 12 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 0 & 0 & 3 & 0 & 0 \\ 0 & 3 & 2 & 6 & 0 \\ 3 & 3 & 0 & 12 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 11 \\ 10 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 2 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix} + c_{4}\begin{bmatrix} 2 \\ 11 \\ 10 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 0 & 0 & 2 & 2 & 0 \\ 2 & 2 & 3 & 11 & 0 \\ 1 & 3 & 2 & 10 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 1 & 1 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix} + c_{4}\begin{bmatrix} 2 \\ 3 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 2 & 1 & 0 & 2 & 0 \\ 4 & 2 & 2 & 3 & 0 \\ 6 & 3 & 1 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0.50 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{2}$ is a free variable, so $c_{2}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 4 \\ 9 \\ 9 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 4 \\ 9 \\ 9 \end{bmatrix} + c_{4}\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 0 & 0 & 4 & 2 & 0 \\ 3 & 2 & 9 & 1 & 0 \\ 1 & 1 & 9 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & -0.50 & 0 \\ 0 & 1 & 0 & -1 & 0 \\ 0 & 0 & 1 & 0.50 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} + c_{4}\begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 1 & 1 & 1 & 3 & 0 \\ 2 & 3 & 2 & 0 & 0 \\ 2 & 1 & 2 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 10 \\ 6 \\ 11 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 10 \\ 6 \\ 11 \end{bmatrix} + c_{4}\begin{bmatrix} 3 \\ 2 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 1 & 3 & 10 & 3 & 0 \\ 2 & 1 & 6 & 2 & 0 \\ 3 & 3 & 11 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & -2 & 0 \\ 0 & 0 & 1 & 1 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 3 \\ 3 \end{bmatrix} + c_{4}\begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 1 & 1 & 0 & 1 & 0 \\ 0 & 3 & 3 & 0 & 0 \\ 3 & 0 & 3 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + c_{4}\begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 2 & 1 & 1 & 2 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 3 & 1 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 7 \\ 5 \\ 9 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} + c_{4}\begin{bmatrix} 7 \\ 5 \\ 9 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 1 & 1 & 2 & 7 & 0 \\ 1 & 0 & 2 & 5 & 0 \\ 1 & 1 & 3 & 9 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 1 & 2 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 6 \\ 12 \\ 8 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 6 \\ 12 \\ 8 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} + c_{4}\begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=0$.
In augmented matrix form: $\left[\begin{array}{cccc|c} 6 & 2 & 0 & 1 & 0 \\ 12 & 3 & 1 & 3 & 0 \\ 8 & 2 & 0 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cccc|c} 1 & 0 & 0 & 0.50 & 0 \\ 0 & 1 & 0 & -1 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{array}\right]$.
Since the number of vectors (4) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 10 \\ 9 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 10 \\ 9 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix} + c_{4}\begin{bmatrix} 1 \\ 3 \\ 3 \end{bmatrix} + c_{5}\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 3 & 0 & 0 & 1 & 1 & 0 \\ 10 & 0 & 2 & 3 & 2 & 0 \\ 9 & 3 & 0 & 3 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0.33 & 0.33 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -0.17 & -0.67 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} + c_{4}\begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix} + c_{5}\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 3 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 2 & 2 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0.33 & 0.33 & 0 \\ 0 & 1 & 0 & 2 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 6 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 6 \\ 2 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix} + c_{4}\begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix} + c_{5}\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 6 & 2 & 2 & 0 & 1 & 0 \\ 2 & 1 & 0 & 0 & 1 & 0 \\ 2 & 3 & 0 & 3 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & -0.75 & 0.50 & 0 \\ 0 & 1 & 0 & 1.50 & 0 & 0 \\ 0 & 0 & 1 & 0.75 & -1 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 8 \\ 7 \\ 8 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} + c_{4}\begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} + c_{5}\begin{bmatrix} 8 \\ 7 \\ 8 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 2 & 1 & 2 & 2 & 8 & 0 \\ 3 & 0 & 2 & 1 & 7 & 0 \\ 1 & 2 & 3 & 2 & 8 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 1 & 3 & 0 \\ 0 & 1 & 0 & 2 & 4 & 0 \\ 0 & 0 & 1 & -1 & -1 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 8 \\ 12 \\ 12 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 3 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} + c_{4}\begin{bmatrix} 8 \\ 12 \\ 12 \end{bmatrix} + c_{5}\begin{bmatrix} 1 \\ 3 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 3 & 3 & 1 & 8 & 1 & 0 \\ 3 & 1 & 0 & 12 & 3 & 0 \\ 3 & 3 & 3 & 12 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 5 & 1.50 & 0 \\ 0 & 1 & 0 & -3 & -1.50 & 0 \\ 0 & 0 & 1 & 2 & 1 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 6 \\ 7 \\ 6 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} + c_{4}\begin{bmatrix} 6 \\ 7 \\ 6 \end{bmatrix} + c_{5}\begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 2 & 2 & 1 & 6 & 2 & 0 \\ 1 & 3 & 0 & 7 & 3 & 0 \\ 2 & 0 & 3 & 6 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & -1 & -1 & 0 \\ 0 & 1 & 0 & 2.67 & 1.33 & 0 \\ 0 & 0 & 1 & 2.67 & 1.33 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 4 \\ 4 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 0 \\ 3 \end{bmatrix} + c_{4}\begin{bmatrix} 2 \\ 4 \\ 4 \end{bmatrix} + c_{5}\begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 0 & 0 & 2 & 2 & 2 & 0 \\ 2 & 3 & 0 & 4 & 2 & 0 \\ 1 & 1 & 3 & 4 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & -1 & -2 & 0 \\ 0 & 1 & 0 & 2 & 2 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 4 \\ 4 \\ 8 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} + c_{4}\begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix} + c_{5}\begin{bmatrix} 4 \\ 4 \\ 8 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 2 & 1 & 2 & 3 & 4 & 0 \\ 0 & 1 & 2 & 1 & 4 & 0 \\ 1 & 3 & 2 & 1 & 8 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & -0.50 & 2 & 0 \\ 0 & 0 & 1 & 0.75 & 1 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} + c_{4}\begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} + c_{5}\begin{bmatrix} 2 \\ 2 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 1 & 2 & 0 & 2 & 2 & 0 \\ 3 & 3 & 1 & 1 & 2 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & -2 & -1 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 4 \\ 6 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 2 \\ 0 \end{bmatrix} + c_{4}\begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} + c_{5}\begin{bmatrix} 4 \\ 6 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 1 & 3 & 2 & 0 & 4 & 0 \\ 2 & 3 & 2 & 1 & 6 & 0 \\ 1 & 2 & 0 & 2 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 1 & 2 & 0 \\ 0 & 1 & 0 & 0.50 & 0 & 0 \\ 0 & 0 & 1 & -1.25 & 1 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 5 \\ 5 \\ 6 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 3 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 1 \\ 0 \end{bmatrix} + c_{4}\begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix} + c_{5}\begin{bmatrix} 5 \\ 5 \\ 6 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 0 & 2 & 3 & 2 & 5 & 0 \\ 3 & 1 & 1 & 0 & 5 & 0 \\ 3 & 3 & 0 & 0 & 6 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & -0.25 & 1 & 0 \\ 0 & 1 & 0 & 0.25 & 1 & 0 \\ 0 & 0 & 1 & 0.50 & 1 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 6 \\ 4 \\ 8 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 3 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} + c_{4}\begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix} + c_{5}\begin{bmatrix} 6 \\ 4 \\ 8 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 2 & 2 & 1 & 0 & 6 & 0 \\ 3 & 1 & 1 & 0 & 4 & 0 \\ 3 & 1 & 1 & 2 & 8 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0.25 & 0 & 0.50 & 0 \\ 0 & 1 & 0.25 & 0 & 2.50 & 0 \\ 0 & 0 & 0 & 1 & 2 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 4 \\ 6 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 2 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix} + c_{4}\begin{bmatrix} 3 \\ 0 \\ 3 \end{bmatrix} + c_{5}\begin{bmatrix} 4 \\ 6 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 2 & 0 & 3 & 3 & 4 & 0 \\ 2 & 2 & 1 & 0 & 6 & 0 \\ 1 & 0 & 1 & 3 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 6 & 2 & 0 \\ 0 & 1 & 0 & -4.50 & 1 & 0 \\ 0 & 0 & 1 & -3 & 0 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 7 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix} + c_{4}\begin{bmatrix} 3 \\ 7 \\ 2 \end{bmatrix} + c_{5}\begin{bmatrix} 3 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 3 & 1 & 0 & 3 & 3 & 0 \\ 3 & 3 & 2 & 7 & 1 & 0 \\ 0 & 0 & 1 & 2 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 1 & 1.33 & 0 \\ 0 & 1 & 0 & 0 & -1 & 0 \\ 0 & 0 & 1 & 2 & 0 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 6 \\ 7 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 2 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 6 \\ 7 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix} + c_{4}\begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix} + c_{5}\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 2 & 6 & 3 & 3 & 0 & 0 \\ 2 & 7 & 3 & 1 & 1 & 0 \\ 1 & 3 & 1 & 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 6 & 0 & 0 \\ 0 & 1 & 0 & -2 & 1 & 0 \\ 0 & 0 & 1 & 1 & -2 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} + c_{4}\begin{bmatrix} 0 \\ 3 \\ 0 \end{bmatrix} + c_{5}\begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 2 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 3 & 2 & 0 \\ 3 & 2 & 0 & 0 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 3 & 2 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 6 \\ 0 \\ 7 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 6 \\ 0 \\ 7 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 0 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix} + c_{4}\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} + c_{5}\begin{bmatrix} 2 \\ 0 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 6 & 2 & 0 & 2 & 2 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 7 & 3 & 3 & 3 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & -1.50 & 0 & 0.50 & 0 \\ 0 & 1 & 4.50 & 0 & -0.50 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 8 \\ 10 \\ 10 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 8 \\ 10 \\ 10 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 3 \\ 3 \end{bmatrix} + c_{4}\begin{bmatrix} 0 \\ 2 \\ 2 \end{bmatrix} + c_{5}\begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 8 & 2 & 3 & 0 & 3 & 0 \\ 10 & 2 & 3 & 2 & 3 & 0 \\ 10 & 2 & 3 & 2 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1.50 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{3}$ is a free variable, so $c_{3}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 5 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} + c_{4}\begin{bmatrix} 5 \\ 2 \\ 2 \end{bmatrix} + c_{5}\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 3 & 3 & 1 & 5 & 2 & 0 \\ 0 & 1 & 1 & 2 & 3 & 0 \\ 0 & 3 & 1 & 2 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 1 & 0.33 & 0 \\ 0 & 1 & 0 & 0 & -1 & 0 \\ 0 & 0 & 1 & 2 & 4 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 6 \\ 2 \\ 7 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 6 \\ 2 \\ 7 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 2 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} + c_{4}\begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix} + c_{5}\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=0$.
In augmented matrix form: $\left[\begin{array}{ccccc|c} 6 & 0 & 1 & 2 & 2 & 0 \\ 2 & 2 & 2 & 0 & 1 & 0 \\ 7 & 2 & 1 & 1 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0.27 & 0.36 & 0 \\ 0 & 1 & 0 & -0.64 & 0.32 & 0 \\ 0 & 0 & 1 & 0.36 & -0.18 & 0 \end{array}\right]$.
Since the number of vectors (5) is greater than the dimension (3), there must be free variables. $c_{4}$ is a free variable, so $c_{4}$ can take a nonzero value, so the vectors are dependent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 2 & 3 & 0 \\ 3 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 3 & 1 & 0 \\ 3 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 0 & 1 & 0 \\ 3 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 2 & 0 & 0 \\ 1 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 2 & 2 & 0 \\ 1 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 2 & 3 & 0 \\ 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 2 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 1 & 0 \\ 1 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 1 & 0 \\ 0 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 0 & 3 & 0 \\ 1 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 3 & 0 \\ 1 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 3 & 0 \\ 0 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 2 & 0 & 0 \\ 3 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 3 & 0 \\ 2 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 1 & 0 \\ 0 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 0 & 2 & 0 \\ 0 & 3 & 0 \\ 1 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 2 & 0 \\ 2 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 3 & 0 \\ 0 & 0 & 0 \\ 0 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 2 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 2 & 2 & 0 \\ 2 & 2 & 0 \\ 1 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 3 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 2 & 0 \\ 2 & 3 & 0 \\ 3 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 3 & 3 & 0 \\ 0 & 1 & 0 \\ 2 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 0 & 2 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 3 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 1 & 0 \\ 3 & 1 & 0 \\ 3 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 3 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 3 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 0 & 3 & 0 \\ 3 & 3 & 0 \\ 2 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 1 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 2 & 1 & 0 \\ 1 & 0 & 0 \\ 3 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 0 & 0 & 0 \\ 1 & 3 & 0 \\ 1 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 2 & 3 & 0 \\ 0 & 3 & 0 \\ 1 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 0 & 0 & 0 \\ 1 & 3 & 0 \\ 2 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 2 & 1 & 0 \\ 1 & 3 & 0 \\ 3 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 0 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=0$.
In augmented matrix form: $\left[\begin{array}{cc|c} 2 & 0 & 0 \\ 0 & 0 & 0 \\ 2 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 0 & 1 & 0 \\ 1 & 0 & 2 & 0 \\ 1 & 2 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 2 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 2 & 3 & 0 \\ 2 & 0 & 1 & 0 \\ 2 & 0 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 0 & 1 & 0 \\ 2 & 2 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 0 & 2 & 0 \\ 3 & 1 & 1 & 0 \\ 1 & 1 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 2 & 0 & 0 \\ 1 & 1 & 2 & 0 \\ 3 & 1 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 3 & 2 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 0 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 1 & 1 & 0 \\ 2 & 1 & 3 & 0 \\ 2 & 0 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 0 & 1 & 0 \\ 3 & 1 & 0 & 0 \\ 1 & 1 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 2 & 2 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 1 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 3 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 2 & 2 & 0 \\ 1 & 2 & 3 & 0 \\ 0 & 3 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 2 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 2 & 1 & 0 \\ 2 & 2 & 1 & 0 \\ 3 & 3 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 1 & 1 & 0 \\ 3 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 2 & 2 & 0 \\ 1 & 0 & 3 & 0 \\ 2 & 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 1 & 2 & 0 \\ 0 & 2 & 3 & 0 \\ 1 & 3 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 1 & 1 & 0 \\ 2 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 0 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 2 & 3 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 2 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 1 \\ 3 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 1 \\ 3 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 0 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 1 & 2 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 3 & 1 & 0 \\ 0 & 2 & 3 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 3 \\ 2 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 2 & 1 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 2 & 1 & 0 \\ 2 & 1 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 0 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 3 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 0 & 3 & 0 \\ 1 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 1 \\ 1 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \\ 1 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 1 \\ 0 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 2 & 2 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 1 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 1 \\ 2 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 0 \\ 1 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 2 & 2 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 3 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 3 \\ 2 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \\ 2 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 3 & 0 & 0 \\ 3 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 \\ 2 & 0 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 3 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 0 \\ 2 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 0 \\ 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 1 & 3 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 2 & 3 & 0 \\ 0 & 1 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 2 \\ 1 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 3 \\ 2 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 2 & 0 & 0 \\ 2 & 3 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 3 & 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 0 \\ 0 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 0 \\ 2 \\ 2 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 2 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 2 \\ 0 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 1 & 1 & 0 \\ 0 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 0 \\ 2 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 2 \\ 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 3 \\ 0 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 1 & 3 & 0 \\ 2 & 0 & 3 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 3 \\ 3 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 3 \\ 3 \\ 2 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 0 \\ 1 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 3 & 1 & 0 & 0 \\ 3 & 0 & 1 & 0 \\ 2 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 1 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 1 \\ 2 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 1 & 3 & 0 \\ 1 & 1 & 0 & 0 \\ 2 & 1 & 1 & 0 \\ 2 & 0 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 2 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 3 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 2 \\ 0 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \\ 2 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 2 \\ 2 \\ 3 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 3 & 2 & 0 \\ 2 & 1 & 2 & 0 \\ 0 & 2 & 3 & 0 \\ 1 & 1 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 3 \\ 1 \\ 2 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 3 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 0 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix} + c_{2}\begin{bmatrix} 0 \\ 2 \\ 0 \\ 3 \end{bmatrix} + c_{3}\begin{bmatrix} 3 \\ 0 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 0 & 0 & 3 & 0 \\ 0 & 2 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 3 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 1 \\ 3 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 1 \\ 3 \\ 3 \\ 2 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 1 \\ 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 1 & 1 & 1 & 0 \\ 3 & 0 & 1 & 0 \\ 3 & 0 & 0 & 0 \\ 2 & 1 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 0 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 0 \\ 2 \\ 3 \end{bmatrix} + c_{2}\begin{bmatrix} 2 \\ 0 \\ 1 \\ 1 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 2 & 1 & 1 & 0 \\ 3 & 1 & 0 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.

Q: Are the following vectors linearly independent? $\left\{ \begin{bmatrix} 2 \\ 2 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: The vectors are independent if the only solution to $c_{1}\begin{bmatrix} 2 \\ 2 \\ 1 \\ 0 \end{bmatrix} + c_{2}\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix} + c_{3}\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ is $c_{1}=c_{2}=c_{3}=0$.
In augmented matrix form: $\left[\begin{array}{ccc|c} 2 & 1 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$, which row reduces to $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$.
The only solution is $c_{1}=c_{2}=c_{3}=0$, so the vectors are independent.