Volume of linear transformation of Jordan domain

ianchenmu

Member
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 $e_1,...,e_n$ 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 $e_{\pi(i)}$, $i=1,...,n$, where $\pi$ is a permutation of the set $\left \{ 1,...,n \right \}$. If $T(x)=Ax$, then $Vol(T(R))=|R|$.

(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 $Vol(T(R))=|R|$.

(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 $E\subset\mathbb{R}^n$, $Vol(T(E))=|det(A)|Vol(E)$.

(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 $E\subset\mathbb{R}^n$, $Vol(T(E))=|det(A)|Vol(E)$, 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 $Vol(f(E))=|det(A)|Vol(E)$ 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 $Vol(R)$ and $Vol(E)$? Why for rectangle $Vol(T(R))=R$ but for Jordan domain, $Vol(T(E))=|det(A)|Vol(E)$?) Thank you.

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