Suppose that $V$ and $W$ are finite-dimensional vector spaces over $\mathbb{R}$. What does it mean to say that $\psi: V \rightarrow W$ is a linear map? State the rank-nullity formula. Using it, or otherwise, prove that a linear map $\psi: V \rightarrow V$ is surjective if, and only if, it is injective.

Suppose that $\psi: V \rightarrow V$ is a linear map which has a right inverse, that is to say there is a linear map $\phi: V \rightarrow V$ such that $\psi \phi=\mathrm{id}_{V}$, the identity map. Show that $\phi \psi=\mathrm{id}_{V}$.

Suppose that $A$ and $B$ are two $n \times n$ matrices over $\mathbb{R}$ such that $A B=I$. Prove that $B A=I$.