How to compute the volume of a parallelotope

Question

Suppose I have $X_1, ..., X_m$ vectors in $\mathbb{R}^N$, linearly independent. In a paper I am reading it says the $m$-dimensional volume with respect to the metric induced by $\| \cdot \|$ (the Euclidean norm on $\mathbb{R}^N$)of the parallelotope spanned by $\{X_1, ..., X_m\}$. What is meant by this and how can I compute this volume?

Answer 1

I'm not sure how to answer the question of "what is meant by this". Perhaps it would be best to understand what happens in the case where $N = 3$ and $m \leq N$.

In any case, the desired volume can be computed as follows. Take $X$ to be the matrix whose columns are $X_1,\dots,X_m$. Then the volume of the parallelotope will be $$ V = \sqrt{\det(X^TX)}. $$ Note that in the case where $m = N$, this reduces to the familiar formula $V = |\det(X)|$.