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Let $T:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a linear transformation and $R\in \mathbb{R}^n$ be a rectangle. Prove:

(1) Let $e_1,\ldots,e_n$ be the standard basis vectors of $\mathbb{R}^n$ (i.e. the columns of the identity matrix). A permutation matrix $A$ is a matrix whose columns are $e_{\pi(i)}$, $i=1,\ldots,n$, where $\pi$ is a permutation of the set $\left \{ 1,\ldots,n \right \}$. If $T(x)=Ax$, then $\operatorname{Vol}(T(R))=|R|$.

(2) let $A=I+B$ be an $n\times n$ matrix where $B$ has exactly one non-zero entry $s=B_{i,j}$ with $i\neq j$. If $T(x)=Ax$, show that $\operatorname{Vol}(T(R))=|R|$.

(3) Recall that a matrix $A$ is elementary if $A$ is a permutation matrix as in (1), or $A=I+B$ as in (2), or $A$ is diagonal with all but one diagonal entry equal to $1$. Deduce that if $T(x)=Ax$ and $A$ is an elementary matrix, then for any Jordan domain $E\subset\mathbb{R}^n$, Vol$(T(E))=|\det(A)|\operatorname{Vol}(E)$.

(4) Recall from linear algebra (row reduction), that any invertible $n\times n$ matrix $A$ is a product of elementary matrices. Prove that for any Jordan domain $E\subset\mathbb{R}^n$, $\operatorname{Vol}(T(E))=|\det(A)|\operatorname{Vol}(E)$, where $T(x)=Ax$ is invertible.

(5) Is (4) true if we do not assume $T$ is invertible?

(6) Prove: If $f: \mathbb{R}^n\rightarrow\mathbb{R}^n$ is an affine transformation and $E\subset\mathbb{R}^n$ is a Jordan domain, then $\operatorname{Vol}(f(E))=|\det(A)|\operatorname{Vol}(E)$ where $A=Df(x)$ is the derivative of $f$ at some point $x$.

((1) and (2)are easy but I have little ideas about the rest. What's the volume of a Jordan domain and what's the relationship between $\operatorname{Vol}(R)$ and $\operatorname{Vol}(E)$? Why for rectangle $Vol(T(R))=R$ but for Jordan domain, $\operatorname{Vol}(T(E))=|\det(A)|\operatorname{Vol}(E)$?) Thank you.

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For (5), take $T$ any projection for example. The image by $T$ of any Jordan domain will have volume $0$ –  leo Mar 24 '13 at 2:14
    
can you provide a more complete answer? And not only for (5). Thanks! –  Ian Mar 24 '13 at 2:18
    
@leo: These are important for me. So please guide me somehow. Thank you! –  Ian Mar 24 '13 at 3:43
    
@leo: Can you tell me how to compute the volume of a Jordan domain? –  Ian Mar 24 '13 at 3:56
    
@leo: Can you help me with these questions please. Thank you so much! –  Ian Mar 24 '13 at 22:47

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