$n$-volume of a ball is proportional to the $n$-th power of the unit $n$-volume

Question

I am searching for a paper wich demonstrates this proposition so I can reference it in an essay I'm currently working on. In mathematical notation, the proposition looks something like this:

$$V_n(r)=\alpha r^n \quad, \quad \alpha = V_n(1)$$

I've been digging arround internet, but the best I could find was the proof wich the Wikipedia page offers and our techer told us that we couldn't refer to any Wikipedia page. Can someone help, please?

Answer 1

This is simply a result of a property of determinants. Suppose you have a subset $A\subseteq\mathbb{R}^n$ whose volume is, say, $\alpha$. Given $r>0$, what is the volume of the set $rA$? Since $rA$ is obtained from $A$ by applying the linear transformation $Tx= rx$, we have $$V(rA)=V(TA)=|\det T|V(A)=r^nV(A)=r^n\alpha$$

So if you need a reference, just open any linear algebra book and look for the volume formula of determinants.