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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

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    How is $V$ defined?2012-02-29

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