Exercise 14

This is an alternative proof of Lemma 2.9. Given an $n\!\times\!n$ matrix $H$, fix a domain $V$ and codomain $W$ of appropriate dimension $n$, and bases $B,D$ for those spaces, and consider the map $h$ represented by the matrix.

• Show that $h$ is onto if and only if there is at least one ${\rm Rep}_{B}(\vec{v})$ associated by $H$ with each ${\rm Rep}_{D}(\vec{w})$.

• Show that $h$ is one-to-one if and only if there is at most one ${\rm Rep}_{B}(\vec{v})$ associated by $H$ with each ${\rm Rep}_{D}(\vec{w})$.

• Consider the linear system $H\cdot{\rm Rep}_{B}(\vec{v})={\rm Rep}_{D}(\vec{w})$. Show that $H$ is nonsingular if and only if there is exactly one solution ${\rm Rep}_{B}(\vec{v})$ for each ${\rm Rep}_{D}(\vec{w})$.