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 $v_{1},\ldots ,v_{n}$ for $V$ and a basis $w_{1},\ldots ,w_{m}$ for $W$. Then we can get a matrix by setting the $k$th column to be $Tv_{k}$ written in the basis $w_{1},\ldots ,w_{m}$.