Exercise 10

Assume that $h$ is a linear transformation of $V$ and that $\langle\vec{\beta}_{1},\ldots,\vec{\beta}_{n}\rangle$ is a basis of $V$. Prove each statement.

• If $h(\vec{\beta}_{i})=\vec{0}$ for each basis vector then $h$ is the zero map.

• If $h(\vec{\beta}_{i})=\vec{\beta}_{i}$ for each basis vector then $h$ is the identity map.

• If there is a scalar $r$ such that $h(\vec{\beta}_{i})=r\cdot\vec{\beta}_{i}$ for each basis vector then $h(\vec{v})=r\cdot\vec{v}$ for all vectors in $V$.