- #1

JG89

- 728

- 1

Let [tex] a_1, a_2, ..., a_k [/tex] be independent vectors in R^n. We define the k-dimensional parallelopiped [tex] \mathbb{P}(a_1, ..., a_k) [/tex] to be the set of all x in R^n such that [tex] x = c_1a_1 + \cdots + c_k a_k [/tex] for scalars c_i such that 0 <= c_i <= 1.

Now let X be the n by k matrix whose i'th column vector is a_i. We define the k-dimensional volume of the parallelopiped [tex] \mathbb{P}(a_1, ..., a_k) [/tex] to be the number [tex] \sqrt{det[X^{tr}X]} [/tex].

My question is, why do they define the volume-function this way? Where is the geometric motivation?

All help would be greatly appreciated!