Exercise 16

Suppose that $h\colon V\to W$ is one-to-one so that by Theorem 2.20, for any basis $B=\langle\vec{\beta}_{1},\dots,\vec{\beta}_{n}\rangle\subset V$ the image $h(B)=\langle h(\vec{\beta}_{1}),\dots,h(\vec{\beta}_{n})\rangle$ is a basis for $h(V)$. (If $h$ is onto then $h(V)=W$.)

• Represent the map $h$ with respect to $\langle B,h(B)\rangle$.

• For a member $\vec{v}$ of the domain, where the representation of $\vec{v}$ has components $c_{1}$, …, $c_{n}$, represent the image vector $h(\vec{v})$ with respect to the image basis $h(B)$.