Suppose that $V,W$ are finite dimensional linear spaces, and that $A:V\to W$ is a linear map from $V$ to $W$.
A non-negative map $v\mapsto N(v)$ is called a seminorm if $N(av)= |a| N(v)$ for every $a$ real number and every vector $v$ and if $N(u+v)\leq N(u)+N(v)$ for any two vectors $u,v$.
Let $\|\cdot \|_W$ be a norm on $W$, and define
$$\|x\| := \|Ax\|_W.$$
Then it is necessarily TRUE that: