Lower Dimensional Volume under Transformation

Question

It is a well known fact that if $K$ is a measurable set in $R^n$ (we can restrict to convex bodies if you like), and $T$ a linear transformation then $$|TK|=\left|\det T\right||K|.$$ If $K$ is not $n-$dimensional then this equality gives $0=0.$ But $K$ has some $m$-dimensional volume, $m<n$. Does there exist a similar relation for the $m$-dimensional volume of $TK$?

Answer 1

Yes. Replace $|\det T|$ with $\sqrt{ \det(TT^*)}$, which is the square root of the sum of the square of the determinants of the $m \times m$ minors of the matrix of $T$.