\[A=\begin{bmatrix}1&2&1&0&-1\\1&2&2&3&0\\0&0&1&3&1\end{bmatrix}\rightsquigarrow \begin{bmatrix}1&2&0&-3&-2\\0&0&1&3&1\\0&0&0&0&0\end{bmatrix}=\mbox {rref}(A)\]
Find each of the following:
\[\text {rank}(A)=\answer {2}\]
\[\text {dim}\left (\text {row}(A)\right )=\answer {2}\]
\[\text {dim}\left (\text {col}(A)\right )=\answer {2}\]
\[\text {dim}\left (\text {null}(A)\right )=\answer {3}\]
Select an appropriate basis for \(\text {row}(A)\).
\(\left \{ \begin{bmatrix}1&2&1&0&-1\end{bmatrix}, \begin{bmatrix}1&2&2&3&0\end{bmatrix}, \begin{bmatrix}0&0&1&3&1\end{bmatrix}\right \}\) \(\left \{ \begin{bmatrix}1&2&0&-3&-2\end{bmatrix}, \begin{bmatrix}0&0&1&3&1\end{bmatrix}, \begin{bmatrix}0&0&0&0&0\end{bmatrix}\right \}\) \(\left \{ \begin{bmatrix}1&2&0&-3&-2\end{bmatrix}, \begin{bmatrix}0&0&1&3&1\end{bmatrix}\right \}\)
Using the algorithm in Subspaces of \(\RR ^n\) Associated with Matrices, find a basis for \(\text {col}(A)\).
\[\left \{\begin{bmatrix}\answer {1}\\\answer {1}\\\answer {0}\end{bmatrix}, \begin{bmatrix}\answer {1}\\\answer {2}\\\answer {1}\end{bmatrix}\right \}\]