Lemma 13.1.17 (Image of Convex).label Let $c\suf\R^{n}$ be convex and $T$ is linear or affine then $T(c)$ is convex.
Proof. Let $u,v\in T(c)$ then $u=T(x),v=T(y)$ for some $x,y\in c$. Let $\lam\in [0,1]$ then
\begin{align*}\lam u+(1-\lam v)&=\lam T(x)+(1-\lam)T(y)\\&=\lam L(x)+\lam b+(1-\lam)L(y)+(1-\lam)b\\&=\lam L(x)+(1-\lam)L(y)+b\\&=L(\lam x+(1-\lam)y)+b\\&=T(\lam x+(1-\lam)y)\in T(c)\end{align*}
$\square$
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