Let $P = \{x\in \mathbb{R}^n \ | \ Ax\leq b\}$ be a bounded polyhedron. Let $L:\mathbb{R}^n \rightarrow \mathbb{R}^k$ be a linear map. Show that $L(P):=\{L(x)\ | \ x\in P\}$ is a bounded polyhedron.
I understand that polyhedral sets are preserved by images and preimages under linear transformations.
To prove it I need to use the fact that the linear map can be represented as a matrix, but I am still not sure how to prove it.
Since $L$ is a mapping between finite-dimensional vector spaces, it is bounded. That is, there exists a number $\|L\|$ such that $$ \|Lx\|\le \|L\|\cdot \|x\|\quad \forall x\in X. $$ Here, $\|x\|$ is the Euclidean norm of $x$ (could be any other vector norm as well). Since $P$ is bounded, there is $M>0$ such that $\|x\|\le M$ for all $x\in P$. This implies that $$ \|Lx\| \le \|L\|\cdot \|x\| \le \|L\|\cdot M \forall x\in P, $$ and $L(P)$ is bounded.