Prove that $|$ det $g| v(U)$ is the volume of $g(U)$ for any linear transformation $g: \Bbb R^n \to \Bbb R^n$.

Question

This is question of spivak's Calculus of Manifolds;

(a) Let $g: \Bbb R^n \to \Bbb R^n$ be a linear transformation of one of the fol­lowing types : $$ \left\{ \begin{array}\\ g(e_i) = e_i, i\neq j\\ g(e_j) = a e_j \end{array} \right. $$

$$ \left\{ \begin{array}\\g(e_i)= e_i, i \neq j\\ g(e_j)=e_j+e_k\end{array} \right. $$

$$ \left\{ \begin{array}\\g(e_k)=e_k, k \neq i,j\\ g(e_i) = e_j\\ g(e_j) = e_i \end{array} \right. $$ If $U$ is a rectangle, show that the volume of $g(U)$ is $|$det $ g| · v(U)$.

(b) Prove that $|$ det $g| v(U)$ is the volume of $g(U)$ for any linear transformation $g: \Bbb R^n \to \Bbb R^n$.

I am not getting any clue how to prove any part of this for b) I think any linear transformation can be written by composition of linear transformations of the type considered in a). But how to write this as a composition is also a question?

Answer 1

Here is an example for the first part of (a).

Let $U = \prod_{k=1}^n [c_k, d_k] $. Note that $v(U) = \prod_{k=1}^n |c_k -d_k|$.

Then $g(U) = [c_1,d_1] \times \cdots \times [a c_j, a d_j] \times \cdots [c_n,d_n]$ (adjusted appropriately for the corner cases, and note that $a$ may be negative). Then it should be clear that $v(g(U)) = |a|v(U) = |\det g|v(U)$.

The third part of (a) is equally straightforward.

The second part of (a) needs to look at how the $[c_j, d_j] \times [c_k, d_k]$ (adjusted appropriately for order) is changed by $g$. In this case the rectangle is mapped into two congruent triangles whose total area is the same as the original rectangle.