Proving a statement about non-negative vectors

Question

How do you prove the following?

Let $C \in \mathbb{R}^{m \times n}$ be given. Prove that if $C$ is nonnegative then the image of a nonnegative vector in $\mathbb{R}^n$ is a nonnegative vector in $\mathbb{R}^m$ when $C$ is applied.

Answer 1

$Hint$: Use that if $a>0$ and $b>0$ so is $ab>0$ and use the definition of matrix multiplication. (As I assume $\mathbf C$ is a matrix representing a linear transformation).