Title: Do Vecotrs Span Rn

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: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -1 \\ -1 \end{bmatrix}, \begin{bmatrix} 3 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & 3 \\ -1 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -3 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & 2 \\ 3 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -1 \\ -3 \end{bmatrix}, \begin{bmatrix} 0 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & 0 \\ -3 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -1 \\ -2 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & 2 \\ -2 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 0 \\ 2 \end{bmatrix}, \begin{bmatrix} -2 \\ -1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & -2 \\ 2 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -3 \\ -2 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & 1 \\ -2 & 3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} -3 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & -3 \\ 2 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 1 \\ -3 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & -1 \\ -3 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -1 \\ 3 \end{bmatrix}, \begin{bmatrix} -3 \\ 1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & -3 \\ 3 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 3 \\ -2 \end{bmatrix}, \begin{bmatrix} 0 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & 0 \\ -2 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 3 \\ -2 \end{bmatrix}, \begin{bmatrix} 2 \\ -3 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & 2 \\ -2 & -3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 3 \\ -2 \end{bmatrix}, \begin{bmatrix} -3 \\ -1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & -3 \\ -2 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -2 \\ 3 \end{bmatrix}, \begin{bmatrix} -1 \\ 2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & -1 \\ 3 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 2 \\ -2 \end{bmatrix}, \begin{bmatrix} 3 \\ -1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 3 \\ -2 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -1 \\ -2 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & 0 \\ -2 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 2 \\ -1 \end{bmatrix}, \begin{bmatrix} -1 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & -1 \\ -1 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 1 \\ -3 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & 0 \\ -3 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -1 \\ -3 \end{bmatrix}, \begin{bmatrix} -2 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & -2 \\ -3 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 1 \\ -2 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 4 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & 0 & 1 \\ -2 & 3 & 4 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -3 \\ -1 \end{bmatrix}, \begin{bmatrix} 1 \\ -3 \end{bmatrix}, \begin{bmatrix} -8 \\ 4 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & 1 & -8 \\ -1 & -3 & 4 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & -2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -3 \\ -3 \end{bmatrix}, \begin{bmatrix} -2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -5 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & -2 & 1 \\ -3 & 1 & -5 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & -2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -2 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 6 \\ 6 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & 3 & 6 \\ 1 & 3 & 6 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & 1 & 2 \\ 2 & 0 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \begin{bmatrix} 0 \\ 5 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 1 & 0 \\ 3 & -1 & 5 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & -2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \begin{bmatrix} -2 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ -8 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & -2 & 2 \\ 3 & 1 & -8 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & -2 \\ 0 & 1 & -2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 4 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & 0 & 0 \\ 1 & 2 & 4 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -1 \\ 1 \end{bmatrix}, \begin{bmatrix} -2 \\ 3 \end{bmatrix}, \begin{bmatrix} -3 \\ 4 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & -2 & -3 \\ 1 & 3 & 4 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -3 \\ 0 \end{bmatrix}, \begin{bmatrix} -1 \\ 3 \end{bmatrix}, \begin{bmatrix} 7 \\ -3 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & -1 & 7 \\ 0 & 3 & -3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & -2 \\ 0 & 1 & -1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} -2 \\ 1 \end{bmatrix}, \begin{bmatrix} -2 \\ 1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & -2 & -2 \\ 1 & 1 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -1 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ -2 \end{bmatrix}, \begin{bmatrix} 1 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & 3 & 1 \\ 0 & -2 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \begin{bmatrix} -3 \\ -3 \end{bmatrix}, \begin{bmatrix} 2 \\ 6 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & -3 & 2 \\ 3 & -3 & 6 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 0 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ -3 \end{bmatrix}, \begin{bmatrix} -6 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & 0 & -6 \\ 1 & -3 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & -2 \\ 0 & 1 & 0 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 1 \\ -3 \end{bmatrix}, \begin{bmatrix} 3 \\ -1 \end{bmatrix}, \begin{bmatrix} -4 \\ -4 \end{bmatrix}, \begin{bmatrix} 6 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & 3 & -4 & 6 \\ -3 & -1 & -4 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 2 & 0 \\ 0 & 1 & -2 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \begin{bmatrix} -3 \\ -2 \end{bmatrix}, \begin{bmatrix} 7 \\ 3 \end{bmatrix}, \begin{bmatrix} -4 \\ -6 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & -3 & 7 & -4 \\ -1 & -2 & 3 & -6 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 1 & 2 \\ 0 & 1 & -2 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -2 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ -1 \end{bmatrix}, \begin{bmatrix} -4 \\ 6 \end{bmatrix}, \begin{bmatrix} 2 \\ -1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & 2 & -4 & 2 \\ 3 & -1 & 6 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 2 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} -3 \\ -2 \end{bmatrix}, \begin{bmatrix} -6 \\ -2 \end{bmatrix}, \begin{bmatrix} -3 \\ 0 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & -3 & -6 & -3 \\ 1 & -2 & -2 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 2 & 2 \\ 0 & 1 & 2 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 2 \\ -1 \end{bmatrix}, \begin{bmatrix} 2 \\ -2 \end{bmatrix}, \begin{bmatrix} 2 \\ -2 \end{bmatrix}, \begin{bmatrix} 8 \\ -6 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 2 & 2 & 8 \\ -1 & -2 & -2 & -6 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 1 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 3 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ 2 \end{bmatrix}, \begin{bmatrix} -2 \\ -2 \end{bmatrix}, \begin{bmatrix} -1 \\ 3 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & -2 & -2 & -1 \\ -1 & 2 & -2 & 3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & -2 & 1 \\ 0 & 1 & -2 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 3 \\ 1 \end{bmatrix}, \begin{bmatrix} -3 \\ 3 \end{bmatrix}, \begin{bmatrix} -3 \\ -1 \end{bmatrix}, \begin{bmatrix} -6 \\ 6 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & -3 & -3 & -6 \\ 1 & 3 & -1 & 6 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 1 & 0 & 0 \\ 0 & -1 & 2 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & -2 & 0 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -2 \\ 2 \end{bmatrix}, \begin{bmatrix} -1 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ -2 \end{bmatrix}, \begin{bmatrix} -6 \\ 8 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & -1 & 1 & -6 \\ 2 & 2 & -2 & 8 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & -1 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -2 \\ -3 \end{bmatrix}, \begin{bmatrix} -1 \\ -2 \end{bmatrix}, \begin{bmatrix} 4 \\ 7 \end{bmatrix}, \begin{bmatrix} 3 \\ 5 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & -1 & 4 & 3 \\ -3 & -2 & 7 & 5 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & -1 & -1 \\ 0 & 1 & -2 & -1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 2 \\ 2 \end{bmatrix}, \begin{bmatrix} -3 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -3 \end{bmatrix}, \begin{bmatrix} 6 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & -3 & 1 & 6 \\ 2 & 1 & -3 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 1 & -1 & -2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -2 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} -3 \\ -3 \\ -3 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ -3 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & -3 & 1 \\ 3 & -3 & 3 \\ 2 & -3 & -3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ -3 \\ -3 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & -1 & -2 \\ -3 & 0 & 0 \\ -3 & -1 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -3 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ -2 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ 0 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & 2 & 3 \\ 3 & -2 & 2 \\ 0 & 2 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ 1 \\ -3 \end{bmatrix}, \begin{bmatrix} -2 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ -3 \\ -3 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & -2 & 1 \\ 1 & 3 & -3 \\ -3 & 0 & -3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 2 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} -2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 3 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & -2 & 2 \\ 2 & 1 & 3 \\ 0 & 3 & 3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ 1 \\ -3 \end{bmatrix}, \begin{bmatrix} -3 \\ -1 \\ -2 \end{bmatrix}, \begin{bmatrix} -3 \\ 1 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & -3 & -3 \\ 1 & -1 & 1 \\ -3 & -2 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 2 \\ -3 \\ -3 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 3 & 3 \\ -3 & 1 & 1 \\ -3 & 1 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} -1 \\ 3 \\ 1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & 2 & -1 \\ 2 & 0 & 3 \\ 1 & 0 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ -1 \\ -1 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & 3 & 2 \\ -1 & 1 & 0 \\ -1 & -1 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ 3 \\ -2 \end{bmatrix}, \begin{bmatrix} 3 \\ -1 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ -1 \\ 0 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & 3 & -2 \\ 3 & -1 & -1 \\ -2 & -1 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 3 \\ -1 \end{bmatrix}, \begin{bmatrix} 0 \\ -3 \\ -3 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & -1 & 0 \\ 3 & 3 & -3 \\ 1 & -1 & -3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} -2 \\ 2 \\ -1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & 1 & -2 \\ -2 & 3 & 2 \\ 3 & 0 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 3 \\ -3 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ -1 \\ -3 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ -3 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & 3 & 0 \\ -3 & -1 & 2 \\ 1 & -3 & -3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ -2 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 3 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & 3 & 3 \\ 3 & 0 & 0 \\ 0 & -2 & 3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 3 \\ -2 \\ -2 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} -2 \\ -1 \\ 3 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & 0 & -2 \\ -2 & 3 & -1 \\ -2 & 2 & 3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} -3 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} -1 \\ -3 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & -3 & -1 \\ 2 & 1 & -3 \\ 1 & 2 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ -1 \\ -3 \end{bmatrix}, \begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 2 \\ -1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & 2 & -1 \\ -1 & -1 & 2 \\ -3 & 1 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -3 \\ 2 \\ -2 \end{bmatrix}, \begin{bmatrix} 2 \\ -2 \\ -2 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & 2 & 0 \\ 2 & -2 & 2 \\ -2 & -2 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -2 \\ -3 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & 3 & 3 \\ -3 & 1 & 0 \\ 1 & 0 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -2 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ -3 \\ 0 \end{bmatrix}, \begin{bmatrix} -2 \\ 3 \\ 3 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & 0 & -2 \\ 3 & -3 & 3 \\ 1 & 0 & 3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ -2 \\ -2 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ -1 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 3 \\ 4 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & 3 & 0 & -1 \\ -2 & 2 & -1 & 3 \\ -2 & -1 & 1 & 4 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ -3 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -2 \\ 8 \\ -4 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & 0 & 1 & -2 \\ 1 & -3 & 1 & 8 \\ 1 & 3 & 0 & -4 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 0 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ -2 \\ 2 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} -4 \\ -4 \\ 0 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & -1 & 2 & -4 \\ -2 & 1 & 3 & -4 \\ 2 & -1 & 1 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ -3 \\ 3 \end{bmatrix}, \begin{bmatrix} -1 \\ -3 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ 0 \end{bmatrix}, \begin{bmatrix} -2 \\ -6 \\ 12 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & -1 & 0 & -2 \\ -3 & -3 & 3 & -6 \\ 3 & 3 & 0 & 12 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -3 \\ -1 \\ -2 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -3 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & -3 & 1 & 1 \\ 1 & -1 & 2 & -3 \\ 0 & -2 & 1 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ -2 \\ -3 \end{bmatrix}, \begin{bmatrix} -2 \\ -3 \\ 3 \end{bmatrix}, \begin{bmatrix} -2 \\ -3 \\ -3 \end{bmatrix}, \begin{bmatrix} 0 \\ -2 \\ 9 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & -2 & -2 & 0 \\ -2 & -3 & -3 & -2 \\ -3 & 3 & -3 & 9 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 3 \\ 3 \\ -2 \end{bmatrix}, \begin{bmatrix} 3 \\ -1 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ -3 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \\ 4 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & 3 & 0 & 0 \\ 3 & -1 & -3 & -1 \\ -2 & 2 & 0 & 4 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \\ -1 \end{bmatrix}, \begin{bmatrix} -1 \\ -3 \\ -3 \end{bmatrix}, \begin{bmatrix} 1 \\ 5 \\ 5 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 0 & -1 & 1 \\ -2 & -1 & -3 & 5 \\ 1 & -1 & -3 & 5 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -2 \\ 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ -2 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ -3 \end{bmatrix}, \begin{bmatrix} -10 \\ 10 \\ 0 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & 3 & 1 & -10 \\ 3 & -2 & 3 & 10 \\ 3 & 3 & -3 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 0 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -3 \\ -3 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \\ -3 \end{bmatrix}, \begin{bmatrix} -2 \\ 2 \\ -1 \end{bmatrix}, \begin{bmatrix} -3 \\ -3 \\ 1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & -1 & -2 & -3 \\ -3 & 1 & 2 & -3 \\ 1 & -3 & -1 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ -2 \\ 1 \end{bmatrix}, \begin{bmatrix} -3 \\ -1 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ -2 \end{bmatrix}, \begin{bmatrix} -9 \\ -7 \\ 8 \end{bmatrix}, \begin{bmatrix} 0 \\ 6 \\ 0 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & -3 & 3 & -9 & 0 \\ -2 & -1 & 2 & -7 & 6 \\ 1 & 3 & -2 & 8 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 1 & -2 \\ 0 & 1 & 0 & 1 & 2 \\ 0 & 0 & 1 & -2 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -2 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} -2 \\ 0 \\ -1 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \\ -3 \end{bmatrix}, \begin{bmatrix} 4 \\ -4 \\ -4 \end{bmatrix}, \begin{bmatrix} 4 \\ -4 \\ -4 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & -2 & -1 & 4 & 4 \\ 2 & 0 & 0 & -4 & -4 \\ 2 & -1 & -3 & -4 & -4 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & -2 & -2 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 2 \\ -3 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \\ -3 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \\ -1 \end{bmatrix}, \begin{bmatrix} 5 \\ -4 \\ 0 \end{bmatrix}, \begin{bmatrix} -2 \\ 0 \\ -2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 1 & -1 & 5 & -2 \\ -3 & -1 & 0 & -4 & 0 \\ 1 & -3 & -1 & 0 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & -2 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ -1 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} -2 \\ -3 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ -8 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & 2 & 0 & -2 & 0 \\ -1 & 0 & 1 & -3 & 2 \\ 3 & 2 & 3 & 3 & -8 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 2 & -2 \\ 0 & 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & -1 & 0 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -2 \\ 1 \\ -2 \end{bmatrix}, \begin{bmatrix} 3 \\ -3 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ -1 \\ 3 \end{bmatrix}, \begin{bmatrix} -1 \\ 4 \\ 5 \end{bmatrix}, \begin{bmatrix} -2 \\ 3 \\ 1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & 3 & 3 & -1 & -2 \\ 1 & -3 & -1 & 4 & 3 \\ -2 & 0 & 3 & 5 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & -1 & 1 \\ 0 & 1 & 0 & -2 & -1 \\ 0 & 0 & 1 & 1 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} -3 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} -4 \\ -4 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ 4 \\ 10 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & 2 & -3 & -4 & -2 \\ 2 & 2 & 0 & -4 & 4 \\ 0 & 3 & 2 & -1 & 10 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 & 2 \\ 0 & 0 & 1 & 1 & 2 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 1 \\ 3 \\ -2 \end{bmatrix}, \begin{bmatrix} -1 \\ 3 \\ -1 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 9 \\ -5 \end{bmatrix}, \begin{bmatrix} 2 \\ 6 \\ -4 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & -1 & 3 & 1 & 2 \\ 3 & 3 & 0 & 9 & 6 \\ -2 & -1 & 1 & -5 & -4 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 2 & 2 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ -3 \\ -2 \end{bmatrix}, \begin{bmatrix} 0 \\ -2 \\ -1 \end{bmatrix}, \begin{bmatrix} -3 \\ -3 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ -1 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & 0 & -3 & 0 & 3 \\ -3 & -2 & -3 & 2 & 2 \\ -2 & -1 & 0 & 2 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & -2 & 1 \\ 0 & 1 & 0 & 2 & -1 \\ 0 & 0 & 1 & 0 & -1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -3 \\ -1 \\ -1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ -2 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix} \right\}$
A: Yes, the set of vectors spans $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & 0 & 1 & 1 & 1 \\ -1 & 0 & -1 & 3 & -1 \\ -1 & -2 & 0 & 5 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & -2 & -1 \\ 0 & 0 & 1 & -2 & 1 \end{bmatrix}$. Since there is a pivot in every row, the set of vectors spans $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -1 \\ 1 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. Since the vectors are in $\mathbb{R}^2$ and there are only 1 vector, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 \\ 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$. There cannot be pivot in every row because there are not enough vectors.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -1 \\ 2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. Since the vectors are in $\mathbb{R}^2$ and there are only 1 vector, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 \\ 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$. There cannot be pivot in every row because there are not enough vectors.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 6 \\ 4 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & 6 \\ 2 & 4 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 \\ 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -2 \\ 1 \end{bmatrix}, \begin{bmatrix} -10 \\ 5 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & -10 \\ 1 & 5 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 5 \\ 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 5 \\ -16 \end{bmatrix}, \begin{bmatrix} 15 \\ -48 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 5 & 15 \\ -16 & -48 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 0 \\ -2 \end{bmatrix}, \begin{bmatrix} 0 \\ -4 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & 0 \\ -2 & -4 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 \\ 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 2 \\ -3 \end{bmatrix}, \begin{bmatrix} 4 \\ -6 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 4 \\ -3 & -6 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 \\ 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -3 \\ -1 \end{bmatrix}, \begin{bmatrix} -6 \\ -2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & -6 \\ -1 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 \\ 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -1 \\ -1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & 0 \\ -1 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -3 \\ -3 \end{bmatrix}, \begin{bmatrix} -1 \\ -1 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & -1 \\ -3 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0.33 \\ 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -3 \\ 1 \end{bmatrix}, \begin{bmatrix} -6 \\ 2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & -6 \\ 1 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 \\ 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -2 \\ 3 \end{bmatrix}, \begin{bmatrix} -4 \\ 6 \end{bmatrix}, \begin{bmatrix} -10 \\ 15 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & -4 & -10 \\ 3 & 6 & 15 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 & 5 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 6 \end{bmatrix}, \begin{bmatrix} 0 \\ 12 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & 0 & 0 \\ 3 & 6 & 12 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 & 4 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 3 \\ 0 \end{bmatrix}, \begin{bmatrix} 6 \\ 0 \end{bmatrix}, \begin{bmatrix} 12 \\ 0 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & 6 & 12 \\ 0 & 0 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 & 4 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 5 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & 0 & 0 \\ 1 & 2 & 5 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 & 5 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -2 \\ 2 \end{bmatrix}, \begin{bmatrix} -6 \\ 6 \end{bmatrix}, \begin{bmatrix} -10 \\ 10 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & -6 & -10 \\ 2 & 6 & 10 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 3 & 5 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 6 \\ 0 \end{bmatrix}, \begin{bmatrix} 10 \\ 0 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 6 & 10 \\ 0 & 0 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 3 & 5 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 1 \\ -3 \end{bmatrix}, \begin{bmatrix} 2 \\ -6 \end{bmatrix}, \begin{bmatrix} 5 \\ -15 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & 2 & 5 \\ -3 & -6 & -15 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 & 5 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 2 \\ -3 \end{bmatrix}, \begin{bmatrix} 4 \\ -6 \end{bmatrix}, \begin{bmatrix} -6 \\ 9 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 4 & -6 \\ -3 & -6 & 9 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 & -3 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} -3 \\ 2 \end{bmatrix}, \begin{bmatrix} -9 \\ 6 \end{bmatrix}, \begin{bmatrix} 9 \\ -6 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & -9 & 9 \\ 2 & 6 & -6 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 3 & -3 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^2$? $\left\{ \begin{bmatrix} 0 \\ -1 \end{bmatrix}, \begin{bmatrix} 0 \\ -3 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^2$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & 0 & 0 \\ -1 & -3 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 3 & -2 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(2) does not have a pivot, the set of vectors does not span $\mathbb{R}^2$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 3 \\ 3 \\ 3 \end{bmatrix}, \begin{bmatrix} 9 \\ 9 \\ 9 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. Since the vectors are in $\mathbb{R}^3$ and there are only 2 vectors, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & 9 \\ 3 & 9 \\ 3 & 9 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 3 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$. There cannot be pivot in every row because there are not enough vectors.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -2 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} -4 \\ 0 \\ 6 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. Since the vectors are in $\mathbb{R}^3$ and there are only 2 vectors, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & -4 \\ 0 & 0 \\ 3 & 6 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$. There cannot be pivot in every row because there are not enough vectors.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 1 \\ -2 \\ -2 \end{bmatrix}, \begin{bmatrix} 2 \\ -4 \\ -4 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. Since the vectors are in $\mathbb{R}^3$ and there are only 2 vectors, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & 2 \\ -2 & -4 \\ -2 & -4 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$. There cannot be pivot in every row because there are not enough vectors.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ -1 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ -2 \\ 6 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. Since the vectors are in $\mathbb{R}^3$ and there are only 2 vectors, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & 0 \\ -1 & -2 \\ 3 & 6 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$. There cannot be pivot in every row because there are not enough vectors.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 4 \\ 2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. Since the vectors are in $\mathbb{R}^3$ and there are only 2 vectors, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & 0 \\ 2 & 4 \\ 1 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$. There cannot be pivot in every row because there are not enough vectors.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 6 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. Since the vectors are in $\mathbb{R}^3$ and there are only 2 vectors, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 3 & 6 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$. There cannot be pivot in every row because there are not enough vectors.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ -3 \\ -1 \end{bmatrix}, \begin{bmatrix} -3 \\ -9 \\ -3 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. Since the vectors are in $\mathbb{R}^3$ and there are only 2 vectors, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & -3 \\ -3 & -9 \\ -1 & -3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 3 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$. There cannot be pivot in every row because there are not enough vectors.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -3 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} -9 \\ 0 \\ 0 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. Since the vectors are in $\mathbb{R}^3$ and there are only 2 vectors, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -3 & -9 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 3 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$. There cannot be pivot in every row because there are not enough vectors.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 3 \\ 2 \\ -1 \end{bmatrix}, \begin{bmatrix} 6 \\ 4 \\ -2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. Since the vectors are in $\mathbb{R}^3$ and there are only 2 vectors, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 3 & 6 \\ 2 & 4 \\ -1 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$. There cannot be pivot in every row because there are not enough vectors.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ -2 \\ 2 \end{bmatrix}, \begin{bmatrix} -1 \\ -4 \\ 2 \end{bmatrix}, \begin{bmatrix} -1 \\ -5 \\ 2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & -1 & -1 \\ -2 & -4 & -5 \\ 2 & 2 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & -0.50 \\ 0 & 1 & 1.50 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & 1 & 2 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 1 \\ -2 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ -4 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ -3 \\ 0 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & 1 & 1 \\ -2 & -4 & -3 \\ 0 & 0 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0.50 \\ 0 & 1 & 0.50 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ -1 \\ 2 \end{bmatrix}, \begin{bmatrix} -2 \\ -1 \\ 3 \end{bmatrix}, \begin{bmatrix} -1 \\ -1 \\ 2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & -2 & -1 \\ -1 & -1 & -1 \\ 2 & 3 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 3 & 3 \\ 1 & 2 & 2 \\ 2 & 1 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ -1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \\ -2 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & 1 & 1 \\ -1 & -1 & 0 \\ 0 & -2 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & -0.50 \\ 0 & 1 & 0.50 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 2 \\ 0 \\ -2 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 1 & 0 \\ 0 & 0 & 1 \\ -2 & -1 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0.50 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ -2 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \\ -1 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & 0 & -1 \\ 1 & 2 & 1 \\ -1 & -2 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ 2 \\ -1 \end{bmatrix}, \begin{bmatrix} -1 \\ 4 \\ 0 \end{bmatrix}, \begin{bmatrix} -1 \\ 4 \\ 0 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & -1 & -1 \\ 2 & 4 & 4 \\ -1 & 0 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ -2 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ -1 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & 2 & 2 \\ 1 & 2 & 3 \\ -1 & -2 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 2 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 4 \\ -1 \end{bmatrix}, \begin{bmatrix} 4 \\ 4 \\ 0 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 3 & 4 \\ 2 & 4 & 4 \\ 0 & -1 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ -2 \\ 2 \end{bmatrix}, \begin{bmatrix} -2 \\ -4 \\ 4 \end{bmatrix}, \begin{bmatrix} -1 \\ -3 \\ 3 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & -2 & -1 \\ -2 & -4 & -3 \\ 2 & 4 & 3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0.50 \\ 0 & 1 & 0.50 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & -1 & 3 \\ 0 & 0 & 0 \\ 0 & 2 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 3.50 \\ 0 & 1 & 0.50 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 1 \\ 2 \\ -2 \end{bmatrix}, \begin{bmatrix} -1 \\ 4 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ -1 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & -1 & 0 \\ 2 & 4 & 3 \\ -2 & 0 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0.50 \\ 0 & 1 & 0.50 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & -1 & 2 \\ 0 & 0 & 0 \\ 2 & 3 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 4 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -2 \\ 0 \\ -1 \end{bmatrix}, \begin{bmatrix} -3 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} -2 \\ 0 \\ 0 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & -3 & -2 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0.67 \\ 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -2 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -4 \\ 4 \\ 2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & -2 & -1 & -4 \\ 1 & 2 & 1 & 4 \\ 0 & 1 & 0 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ -1 \\ -1 \end{bmatrix}, \begin{bmatrix} 1 \\ -2 \\ -2 \end{bmatrix}, \begin{bmatrix} 2 \\ -1 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ -2 \\ -2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & 1 & 2 & -2 \\ -1 & -2 & -1 & -2 \\ -1 & -2 & -1 & -2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & -3 & 6 \\ 0 & 1 & 2 & -2 \\ 0 & 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix}, \begin{bmatrix} 2 \\ -4 \\ -2 \end{bmatrix}, \begin{bmatrix} 0 \\ -3 \\ -1 \end{bmatrix}, \begin{bmatrix} -1 \\ 5 \\ 2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & 2 & 0 & -1 \\ -2 & -4 & -3 & 5 \\ -1 & -2 & -1 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ -2 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ -6 \\ 3 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & 1 & 0 & 3 \\ -2 & -2 & -2 & -6 \\ 1 & 1 & 1 & 3 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 1 & 0 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ -1 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 0 & 2 & 1 & -1 \\ 2 & 2 & 2 & 0 \\ 1 & -1 & 0 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0.50 & 0.50 \\ 0 & 1 & 0.50 & -0.50 \\ 0 & 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -2 \\ -2 \\ 2 \end{bmatrix}, \begin{bmatrix} -4 \\ -4 \\ 4 \end{bmatrix}, \begin{bmatrix} -2 \\ -4 \\ 3 \end{bmatrix}, \begin{bmatrix} 8 \\ 8 \\ -8 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -2 & -4 & -2 & 8 \\ -2 & -4 & -4 & 8 \\ 2 & 4 & 3 & -8 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 2 & 0 & -4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ -3 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ -2 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 1 & 3 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ -1 & -3 & -2 & 1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 3 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ -2 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ -2 \\ 1 \end{bmatrix}, \begin{bmatrix} 10 \\ -7 \\ 2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 3 & 3 & 10 \\ -1 & -2 & -2 & -7 \\ 2 & 1 & 1 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 2 \\ 0 \\ -2 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ -2 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ -2 \end{bmatrix}, \begin{bmatrix} -2 \\ 0 \\ 2 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 2 & 2 & -2 \\ 0 & 0 & 0 & 0 \\ -2 & -2 & -2 & 2 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 1 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$. Since row(2) and row(3) does not have pivots, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 4 \\ 0 \\ -2 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ 0 \\ 0 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} 2 & 4 & 1 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & -2 & -1 & 0 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & -0.50 & -1 \\ 0 & 1 & 0.50 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.

Q: Does the following set of vectors span $\mathbb{R}^3$? $\left\{ \begin{bmatrix} -1 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 4 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 4 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -6 \\ -1 \end{bmatrix} \right\}$
A: No, the set of vectors does not span $\mathbb{R}^3$. This can be verified by putting the vectors in a matrix form. The matrix $\begin{bmatrix} -1 & -1 & -1 & 1 \\ 2 & 4 & 4 & -6 \\ 1 & 1 & 1 & -1 \end{bmatrix}$ reduces to $\begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{bmatrix}$. Since row(3) does not have a pivot, the set of vectors does not span $\mathbb{R}^3$.