Exercise 23

Let $h\colon V\to\mathbb{R}$ be a homomorphism, but not the zero homomorphism. Prove that if $\langle\vec{\beta}_{1},\dots,\vec{\beta}_{n}\rangle$ is a basis for the null space and if $\vec{v}\in V$ is not in the null space then $\langle\vec{v},\vec{\beta}_{1},\dots,\vec{\beta}_{n}\rangle$ is a basis for the entire domain $V$.