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Volume of linear transformations of Jordan domain

  1. Mar 24, 2013 #1
    1. The problem statement, all variables and given/known data

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








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




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




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




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




    (5) Is (4) true if we do not assume [itex]T[/itex] is invertible?




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




    2. Relevant equations

    n/a

    3. The attempt at a solution
    (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 [itex]Vol(R)[/itex] and [itex]Vol(E)[/itex]? Why for rectangle [itex]Vol(T(R))=R[/itex] but for Jordan domain, [itex]Vol(T(E))=|det(A)|Vol(E)[/itex]? Thank you.
     
    Last edited: Mar 24, 2013
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