Matrix of $L(A)=A^{T}$ From $R^{2 \times 2} \rightarrow R^{2 \times 2}$

Question

A bit of trouble with this question:

Find the matrix of the linear transformation $L(A)=A^{T}$ From $R^{2 \times 2} \rightarrow R^{2 \times 2}$ with respect to the basis $\begin{bmatrix} 1&0\\0&0\end{bmatrix},\begin{bmatrix} 0&0\\0&1\end{bmatrix},\begin{bmatrix} 0&1\\1&0\end{bmatrix},\begin{bmatrix} 0&1\\-1&0\end{bmatrix}$.

Do I need to run Gram-Schmidt?

Answer 1

Call the basis-matrices $v_1,v_2,v_3,v_4$. We have $$ L(v_1) = v_1\\ L(v_2) = v_2\\ L(v_3) = v_3\\ L(v_4) = -v_4 $$ It follows that the matrix of $L$ with respect to the basis is $$ \pmatrix{1\\&1\\&&1\\&&&-1} $$