Let $P$ be a polyhedron in $\Bbb R^n$, $a \in \Bbb R^n$ be a vector and $b \in \Bbb R$ be a scalar. Consider the set $Q= \{x \in P | {a^T} x \le b \}$.
(a) Prove that $Q$ is also a polyhedron.
(b) Is it always possible to choose $b$ in such a way so that $Q=P$ ? Explain your answer.
Graphically, I can understand that if we fix a $b$ as scalar then divide it by a half-space, then both half space should still be polyhedron. But how can I explain it in words and equations?
Also need some hints about (b).