Exercise 24

Assume that $h\colon V\to W$ is linear.

• Show that the range space of this map $\{\hskip 1.00006pth(\vec{v})\mid\vec{v}\in V\hskip 1.00006pt\}$ is a subspace of the codomain $W$.

• Show that the null space of this map $\{\hskip 1.00006pt\vec{v}\in V\mid h(\vec{v})=\vec{0}_{W}\hskip 1.00006pt\}$ is a subspace of the domain $V$.

• Show that if $U$ is a subspace of the domain $V$ then its image $\{\hskip 1.00006pth(\vec{u})\mid\vec{u}\in U\hskip 1.00006pt\}$ is a subspace of the codomain $W$. This generalizes the first item.

• Generalize the second item.