How to show that the set of feasible solutions of a linear program forms a convex set

Question

How to show that the set of feasible solutions to the following linear program forms a convex set: Minimize $c^Tx$ subject to $Ax = b$ and $x\geq0$

I found this problem not clear. Should I start with converting problem into canonical format? But constraints are weird.

Answer 1

Suppose $Ax_1=b$ and $x_1 \geq 0$ and $Ax_2 =b$ and $x_2 \geq 0$,

let $\lambda \in [0,1]$,

$$A(\lambda x_1 + (1- \lambda)x_2) = \lambda Ax_1 + (1-\lambda)Ax_2 = \lambda b + (1-\lambda )b = b$$

Also, note that since $\lambda \geq 0$ and $(1-\lambda) \geq 0$,

$$\lambda x_1 + (1-\lambda)x_2 \geq 0$$