Results 1 to 1 of 1

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 07:58 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear Algebra Proof Regarding Linear Transformations
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: Mar 1st 2013, 01:53 AM
  2. Linear Transformations and the General Linear Group
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Dec 26th 2011, 11:50 AM
  3. Basic Linear Algebra - Linear Transformations Help
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: Dec 7th 2010, 04:59 PM
  4. Volume preserving transformations
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Dec 11th 2009, 02:03 PM
  5. Replies: 3
    Last Post: Jun 2nd 2007, 11:08 AM

Search Tags

/mathhelpforum @mathhelpforum