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 $w_{1},\ldots ,w_{m}$.

We would hope that any property we attribute to a linear map is invariant of the matrix we use to represent it. For instance if $T:V\to W$ is called "injective" then it should be injective regardless of what matrix we use. Similarly given any matrix that is injective, any of the possible linear maps that that matrix represents should be injective.

Examples of other properties like this: injective, surjective, bijective, rank, diagonalizable

On the other hand, a property like "the sum of the columns is equal to such and such" is not invariant