User:IssaRice/Linear algebra/Linear transformation vs matrix views

Given an $m\times n$ matrix $A$ we can define a linear map $T:{\mathbf{R}}^n\to {\mathbf{R}}^m$ by $T(x)=Ax$.

Given a linear map $T:V\to W$, it is not immediately possible to get a corresponding matrix. We must choose some basis $v_1,\dots ,v_n$ for $V$ and a basis $w_1,\dots ,w_m$ for $W$. Then we can get a matrix by setting the $k$th column to be $T{v}_k$ written in the basis $w_1,\dots ,w_m$.