Define a linear transformation $T:\mathbb{R}^2 \to \mathbb{R}^2 $ by: $$T\left(\begin{bmatrix}x_1\\x_2\end{bmatrix}\right) = \begin{bmatrix}x_2\\x_1\end{bmatrix}$$
Find the standard matrix of $T$, called $A$, and find the basis for $\mbox{range}(A)$, $\mbox{null}(A)$, and the $\mbox{rank}(A)$.
I know that the $\mbox{range}(A)$ is all of the pivot columns in $A$, and the $\mbox{null}(A)$ is defined by $Ax=0$, and the rank theorem states that the $\mbox{rank}(A)+\mbox{dim}(\mbox{null}(A))$ is the number of columns in $A$, so I feel confident I can solve the second part of the problem if I could find $A$, but I need help finding $A$.
Also sorry for the poor formatting, I still don't know how to format questions efficiently on here.
Best Answer
It seem to me that the matrix is of form \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}.