If we let $T : \mathbb R^2 \to \mathbb R^2$ be a linear transformation. Let $B=\{v_1,v_2\}$ with $ v_1 = \begin{bmatrix} 1 \\1 \end{bmatrix}$ and $ v_2 = \begin{bmatrix} 0 \\1 \end{bmatrix}$ and let $e$ be the standard basis for $R^2$. Suppose $M_{B<-B}(T) = \begin{bmatrix} 1&2 \\3&4 \end{bmatrix}$
I found $M_{B<-e}(id) = \begin{bmatrix} 1&0 \\-1&1 \end{bmatrix}$ but what is $M_{e<-e}(T)$?
Hint: $$ M_{e \leftarrow e}[T] = M_{e \leftarrow B}[id]\;M_{B \leftarrow B}[T] \;M_{B\leftarrow e}[id] = \\ M_{e \leftarrow B}[id]\;M_{B \leftarrow B}[T] \; (M_{e \leftarrow B}[id])^{-1} $$