linear algebra – lower dimensional volume in transformation

it's a well known fact that if K $ is a measurable set in $ R ^ n $ (we can limit to convex bodies if you want), and $ T $ a linear transformation then
$$ | TK | = | detT || K |. $$
Yes K $ is not $ n- $dimensional then this equality gives 0 $ = 0. $ But K $ some $ m $– dimensional volume, $ m <n $. Is there a similar relationship for the $ m $three-dimensional volume of $ TK $?