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Thread: volume of linear transformations of Jordan domain

  1. #1
    Mar 2013
    Tallahassee, FL, USA

    volume of linear transformations of Jordan domain

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

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

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

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

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

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

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

    ((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 $\displaystyle Vol(R)$ and $\displaystyle Vol(E)$? Why for rectangle $\displaystyle Vol(T(R))=R$ but for Jordan domain, $\displaystyle Vol(T(E))=|det(A)|Vol(E)$?) Thank you.
    Last edited by ianchenmu; Mar 23rd 2013 at 06:58 PM.
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