Let $C\subset \mathbb{R}^m$ be a convex set, let $A$ be an $m \times n$ matrix and $b \in \mathbb{R}^m$. Show that $S=\{x \in \mathbb{R}^n | Ax+b \in C\}$ is a convex set.
I can't understand why this set is convex. Take $x,y \in S$ such that $Ax+b \in C$ and $Ay+b \in C$, then it follows that $A((1-λ)+λy)=(1-λ)Ax+λAy$. It would be easy if I can use that $Ax=-b$, because in that case I'd say that $A(1-λ)Ax+λAy=(1-\lambda)(-b)+\lambda(-b)=-b$ and hence convex. Then from the assumption it follows that this set is convex. I am, however, not sure whether I can use that or not. Could anyone give me a hint on this proof?
I think you just need to write it out clearly. $$A[(1-\lambda) x + \lambda y] + b = (1-\lambda)[Ax + b] + \lambda [Ay+b] ∈ C$$ by convexity of $C$. Hence, $(1-\lambda) x + \lambda y$ is in your set.