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

    (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.
    Last edited by ianchenmu; Mar 23rd 2013 at 06:58 PM.
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