\[A=\begin{bmatrix}2 & 1 & -1 & 3\\1 & 0 & 3 & 1\\1 & 1 & -4 & 2\end{bmatrix}\]
Use techniques discussed in Image and Kernel of a Linear Transformation to find the basis for the kernel and the image of the linear transformation, \(T_A\), induced by \(A\).
Basis for \(\mbox {im}(T_A)\): \(\left \{\begin{bmatrix}\answer {2}\\\answer {1}\\\answer {1}\end{bmatrix}, \begin{bmatrix}\answer {1}\\\answer {0}\\\answer {1}\end{bmatrix}\right \}\)
Basis for \(\mbox {ker}(T_A)\): \(\left \{\begin{bmatrix}\answer {-3}\\\answer {7}\\1\\0\end{bmatrix}, \begin{bmatrix}\answer {-1}\\\answer {-1}\\0\\1\end{bmatrix}\right \}\)