Volume of linear transformations of Jordan domain

[ianchenmu]

Homework Statement

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].

Homework Equations

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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.

 

[mighty2000]

Hi there, thank you for your post. It seems like you are struggling with some of the concepts in this problem. Let me try to clarify some things for you.

Firstly, the volume of a Jordan domain is a concept from multivariable calculus. Essentially, it is the measure of the space enclosed by the boundary of the domain. For example, in two dimensions, the volume of a rectangle is simply its length multiplied by its width. In higher dimensions, it becomes more complicated, but the basic idea is the same.

In part (1), we are considering a linear transformation T that is represented by a permutation matrix A. This means that the columns of A are simply reordering the standard basis vectors e_1,...,e_n. Since A is a permutation matrix, its determinant is either 1 or -1. Therefore, the volume of the transformed rectangle T(R) is simply equal to the volume of R multiplied by the determinant of A, which is just |R|.

In part (2), we are considering a transformation T represented by a matrix A=I+B, where B has exactly one non-zero entry. This means that T is essentially just a shear transformation in one direction. Again, the volume of the transformed rectangle T(R) is equal to the volume of R multiplied by the determinant of A, which is just |R|.

In part (3), we are considering an elementary matrix A, which is either a permutation matrix, a matrix of the form I+B, or a diagonal matrix with all but one diagonal entry equal to 1. In this case, we can see that the determinant of A is simply equal to the determinant of the non-identity matrix B, which is just the single non-zero entry s. Therefore, the volume of the transformed Jordan domain T(E) is equal to the volume of E multiplied by the determinant of A, which is just |det(A)|.

In part (4), we are considering any invertible matrix A, which can be decomposed into a product of elementary matrices. Since we have already proven that the volume of a transformed Jordan domain is equal to the volume of the original domain multiplied by the determinant of the transformation matrix, we can simply apply this to each elementary matrix in the decomposition of A. Since each elementary matrix has a determinant of either 1 or -1, the overall determinant of A will be |det(A)|, and thus the volume of the transformed Jordan domain will be