(Theorem) Invertible matrix

The invertible matrix theorem applies to two vector spaces with the same dimension. The theorem is built from facts relating inverses and the Algebra of functions. We begin with the following proposition:

Given a linear map $A:V\to W$, if $A$ is:

• Right invertible, then $A$ is surjective;
• Left invertible, then $A$ is injective;
• Invertible, then $A$ is bijective.

In fact, each of these statements is a biconditional:

• If $A$ is surjective, then $A$ is right invertible.
• If $A$ is injective, then $A$ is left invertible.
• If $A$ is bijective, then $A$ is invertible.

#wip check that these are matched correctly

The invertible matrix theorem follows:

Theorem: Invertible matrix

If $A:V\to W$ is a linear map between two vector spaces of the same dimension, then

$$A\text{ is invertible}⟺A\text{ is injective, so }\text{ker}(A)=\{\vec{0}\}⟺A\text{ is surjective.}$$