1. Vector Norms

Let S be a real and nonsingular matrix, and let ||.|| be any norm on R^n. Define ||.||' by ||x||' = ||Sx||. Show that ||.||' is also a norm on R^n.

I know ||S|| = max ||Sx|| from ||x||=1 but I'm not sure if that applies.

2. You just need to check the axioms for a norm. Let $\displaystyle x,y\in \mathbb R^n$, and $\displaystyle \lambda \in \mathbb R$. Then

Positive definiteness:

$\displaystyle \|x\|'=0\Leftrightarrow \| S\cdot x\| =0$
$\displaystyle \Leftrightarrow S\cdot x=0$ (since $\displaystyle \|\cdot\|$ is a norm)
$\displaystyle \Leftrightarrow x=0$ (since $\displaystyle S$ is nonsingular, and thus invertible)

Positive homogeneity:

$\displaystyle \|\lambda\cdot x\|'=\|S\cdot (\lambda\cdot x)\| = \|\lambda \cdot (S\cdot x)\|$
$\displaystyle =|\lambda|\cdot \|S\cdot x\|$ (for $\displaystyle \|\cdot\|$ is a norm)
$\displaystyle =|\lambda|\cdot \|x\|'$

Triangle inequality:

$\displaystyle \|x+y\|'=\|S\cdot (x+y)\|=\| S\cdot x+S\cdot y\|$
$\displaystyle \leq \|S\cdot x\|+\|S\cdot y\|$ (again, $\displaystyle \|\cdot\|$ is a norm)
$\displaystyle =\|x\|'+\|y\|'$