Volume of linear transformation of Jordan domain

[i_a_n]

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

(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.

 

[jvicens]

Thank you for your post. Allow me to address each of your questions in turn.

(1) and (2) - As you correctly stated, these two parts are relatively straightforward. The key idea here is that a linear transformation $T$ preserves the volume of a shape, as long as it is applied to a rectangle or a Jordan domain. This is because a linear transformation preserves the shape and orientation of the original object, and therefore the volume remains unchanged. This is why $Vol(T(R))=|R|$ for the rectangle $R$ and $Vol(T(R))=|R|$ for the Jordan domain $E$.

(3) - The volume of a Jordan domain is defined as the integral of $1$ over the domain, with respect to the standard Lebesgue measure. In other words, it is the "size" of the domain in terms of the standard unit of measure. The key relationship between $Vol(R)$ and $Vol(E)$ is that $Vol(E)=\int_E 1 d\mu$, where $\mu$ is the standard Lebesgue measure. Therefore, the volume of a Jordan domain is a way of measuring the size of the domain, just like the volume of a rectangle is a way of measuring the size of the rectangle. The reason why $Vol(T(E))=|det(A)|Vol(E)$ is because $T$ is a linear transformation, and as mentioned before, linear transformations preserve the volume of a shape. Since $A$ is an elementary matrix, it can be decomposed into a product of permutation matrices, matrices of the form $I+B$, and diagonal matrices with all but one diagonal entry equal to $1$. Each of these types of matrices corresponds to a specific type of elementary transformation, and these transformations preserve the volume of a shape. Therefore, the overall transformation $T$ preserves the volume of the Jordan domain $E$, and the determinant of $A$ corresponds to the scaling factor of the volume. This is why $Vol(T(E))=|det(A)|Vol(E)$.

(4) - This part follows directly from the fact that any invertible matrix $A$ can be decomposed into a product of elementary matrices. Since each of these elementary matrices preserves the volume of a shape, the overall transformation $T$ preserves the volume of the Jordan domain $E$, and the determinant of $A$ corresponds to the scaling factor of the volume.