Exercise 21

Where $\vec{u},\vec{v}\in\mathbb{R}^{n}$, by definition the line segment connecting them is the set $\ell=\text{$\{\hskip 1.00006ptt\cdot\vec{u}+(1-t)\cdot\vec{v}\mid t\in[0..1]\hskip 1.00006pt\}$}$. Show that the image, under a homomorphism $h$, of the segment between $\vec{u}$ and $\vec{v}$ is the segment between $h(\vec{u})$ and $h(\vec{v})$.

A subset of $\mathbb{R}^{n}$ is convex if, for any two points in that set, the line segment joining them lies entirely in that set. (The inside of a sphere is convex while the skin of a sphere is not.) Prove that linear maps from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ preserve the property of set convexity.