(i) What does it mean to say a set $C \subseteq \mathbb{R}^{n}$ is convex?

(ii) What does it mean to say $z$ is an extreme point of a convex set $C ?$

Let $A$ be an $m \times n$ matrix, where $n>m$. Let $b$ be an $m \times 1$ vector, and let

$$C=\left\{x \in \mathbb{R}^{n}: A x=b, x \geqslant 0\right\}$$

where the inequality is interpreted component-wise.

(iii) Show that $C$ is convex.

(iv) Let $z=\left(z_{1}, \ldots, z_{n}\right)^{T}$ be a point in $C$ with the property that at least $m+1$ indices $i$ are such that $z_{i}>0$. Show that $z$ is not an extreme point of $C$. [Hint: If $r>m$, then any set of $r$ vectors in $\mathbb{R}^{m}$ is linearly dependent.]

(v) Now suppose that every set of $m$ columns of $A$ is linearly independent. Let $z=\left(z_{1}, \ldots, z_{n}\right)^{T}$ be a point in $C$ with the property that at most $m$ indices $i$ are such that $z_{i}>0$. Show that $z$ is an extreme point of $C$.