Change of Variables Clarification

Question

How can we show that $v(L(C)) = |\det DL|v(C)$ for any open cube $C$ an element of $\mathbb{R}^n$ and any linear transformation $L: \mathbb{R}^n \rightarrow \mathbb{R}^n$, without direct applying the change of variables theorem?

Thanks

Answer 1

The most elementary proof of this fact that I have seen is given by Axler in Linear Algebra Done Right, 2nd Ed; see Theorem 10.38. He actually refers to his argument as a "pseudoproof" due to the fact that volume has not been defined rigorously - which you really can't do without some form of measure theory.

For a more advanced proof that does involve measure theory, I refer you to this question I asked when I was working through some of these issues myself awhile back. The question/answers/comments contain specific references and tips for understanding the result in a more general setting.