What condition should an $m\times n$ matrix $A$ meet for the function $x\mapsto\|Ax\|$ to define a vector norm. where $\|\cdot\|$ denotes some vector norm in $\mathbb R^m$?
Should $A$ have full rank? Because if $A$ does not have full rank then the first property of matrix norm is violated. That is my solution. Is it right?
Property 1. and 2. in the definition https://en.wikipedia.org/wiki/Norm_(mathematics)#Definition are always satisfied. Property 3. is satisfied if and only if $Av=0$ implies $v=0$ which is equivalent to say that $m\le n$ and $A$ has full rank.