I am trying to work my way through the proof of the change of variables theorem for Lebesgue integrals. A key lemma in this context is as follows:
If $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear map and $A \subset \mathbb{R}^n$ is Lebesgue measurable then $\lambda(T(A)) = |\det T \ | \cdot \lambda (A)$, where $\lambda(X)$ denotes the Lebesgue measure of $X$.
Can anyone provide a reference for a proof of this lemma that clearly references the facts from linear algebra that are necessary to effect the proof? The source I have for this lemma refers to German texts that I am incapable of reading and I have been unsuccessful at finding an alternate proof.
Added For the Benefit of Future Readers: In addition to the excellent references I received in response to this question, I have managed to find an additional reference that also provides a good proof of this fact: Aliprantis and Burkinshaw's Principles of Real Analysis, Third Edition, Lemma 40.4 pp 389-390.