Corollaries from Matrix representations

A linear map $T:V\to W$ is injective if and only if

If $V$ is finite-dimensional, this is equivalent to $\operatorname{nullity}(T)=0$.

Let $A=[T]_{B_V}^{B_W}$ be the matrix representation of $T:V\to W$ over a scalar field $\mathbb F$, with $\dim(V)=n$ and $\dim(W)=m$. Then

and

Since a linear map is injective exactly when its kernel is trivial and surjective exactly when its image is the codomain, these identities give the coordinate tests $\ker(A)=\{0\}$ and $\operatorname{im}(A)=\mathbb F^m$. Here $\ker(A)$ and $\operatorname{im}(A)$ are sets of coordinate columns; the displayed equalities identify them with the corresponding abstract subspaces through the coordinate maps.

Consequently,

Hence $T$ is injective if and only if $\operatorname{nullity}(A)=0$ and $T$ is surjective if and only if $\operatorname{rank}(A)=m$. If $m=n$, then $T$ is bijective if and only if $A$ is invertible.