co-norm of a linear transformation on R^n

$\displaystyle |\;|$ is a norm on $\displaystyle \mathbb{R}^n$.

Define the co-norm of the linear transformation $\displaystyle T : \mathbb{R}^n\rightarrow\mathbb{R}^n$ to be

$\displaystyle m(T)=inf\left \{ |T(x)| \;\;\;\; s.t.\;|x|=1 \right \}$

Prove that if $\displaystyle T$ is invertible with inverse $\displaystyle S$ then $\displaystyle m(T)=\frac{1}{||S||}$.

(I think probably we need to do something with the norm, but I still can't get it... So thank you.)

Re: prove a linear transformation is invertible on R^n

Hey ianchenmu.

Is it possible to relate the norm to the determinant?