The question
Given a matrix $A_{ m \times n}$ and a compact set $C \in \mathbb{R}^{n}$. the set $B$ defined below is compact?
$$B= \{Ax | x \in C\}$$
What ive tried
I believe that $B$, becaue I could not think about any counterexample, is compact so I need to prove that he is closed and limited. I think i could use that $||Ax|| \leq |A||x|$ to prove that it is a limited set, but, I don't know how to prove that he is closed, can someone please give me any ideas or advices on how to prove that? Any hints or dvices are more than welcome.
Best Answer
The function from $\mathbb{R}^n$ to $\mathbb{R}^m$ defined by $$x\mapsto Ax, \quad \forall x\in \mathbb{R}^n,$$ is clearly continuous. Therefore $B$ is the continuous image of a compact subset $C$ and hence compact.