Exercise 22

Let $V,W,X$ be vector spaces with bases $B,C,D$.

• Suppose that $h\colon V\to W$ is represented with respect to $B,C$ by the matrix $H$. Give the matrix representing the scalar multiple $rh$ (where $r\in\mathbb{R}$) with respect to $B,C$ by expressing it in terms of $H$.

• Suppose that $h,g\colon V\to W$ are represented with respect to $B,C$ by $H$ and $G$. Give the matrix representing $h+g$ with respect to $B,C$ by expressing it in terms of $H$ and $G$.

• Suppose that $h\colon V\to W$ is represented with respect to $B,C$ by $H$ and $g\colon W\to X$ is represented with respect to $C,D$ by $G$. Give the matrix representing $g\mathbin{\circ}h$ with respect to $B,D$ by expressing it in terms of $H$ and $G$.