Solution
= Solution
Construct the <matrix of a linear map> by expressing the images of the basis vectors in that same basis:
$$
Le_j=\sum_{i=1}^2A_{ij}e_i.
$$
Thus the $j$th column is $[Le_j]_B$. For $v=v_1e_1+v_2e_2$, linearity gives
$$
\boxed{[Lv]_B=A[v]_B.}
$$
The column $[Lv]_B$ is a coordinate vector; reconstructing the actual vector requires $Lv=(A[v]_B)_1e_1+(A[v]_B)_2e_2$. This distinguishes the <linear map> from its basis-dependent matrix.