[PROPOSITIONS] Matrix representations

Operations on matrix representations

Let $T_A:\mathbb R^n\to\mathbb R^m$ and $T_B:\mathbb R^n\to\mathbb R^m$ be the standard-coordinate matrix maps

T_A(x)=Ax,\qquad T_B(x)=Bx.

Then

(T_A+T_B)(x)=(A+B)x,\qquad (\omega T_A)(x)=(\omega A)x.

If $T_A:\mathbb R^p\to\mathbb R^m$ and $T_B:\mathbb R^n\to\mathbb R^p$ , then

(T_A\circ T_B)(y)=A(By)=(AB)y.

Thus the matrix of a sum, scalar multiple, or composition is respectively the sum, scalar multiple, or product of the matrices, with the product ordered to match the composition.

[1]:Matrix maps are linear

Bijectivity in equal dimensions

For a linear map $T:V\to W$ between finite-dimensional vector spaces:

if $T$ is bijective, then $\dim(V)=\dim(W)$ ;
if $\dim(V)=\dim(W)$ and $T$ is injective, then $T$ is surjective;
if $\dim(V)=\dim(W)$ and $T$ is surjective, then $T$ is injective.

Indeed, if $T$ is bijective, then its kernel is trivial and its image is all of $W$ . Rank-nullity therefore gives $\dim(V)=\operatorname{rank}(T)=\dim(W)$ . If the dimensions are equal and $T$ is injective, then nullity is $0$ , so rank equals $\dim(V)=\dim(W)$ and $T$ is surjective. The surjective case is dual: rank equals $\dim(W)=\dim(V)$ , so nullity is $0$ and $T$ is injective.

Propositions from Matrix representations

Operations on matrix representations

Bijectivity in equal dimensions

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