- #1

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## Main Question or Discussion Point

Assume that [itex]X^1,X^2,\ldots, X^k[/itex] are vectors in [itex]\mathbb{R}^n[/itex], and [itex]1\leq k\leq n[/itex]. Is there a simple formula for the k-dimensional measure of the generalised "quadrangle" spanned by these vectors?

If [itex]k=n[/itex], then the solution is [itex]|\textrm{det}(X)|[/itex] with [itex]X_{ij}=(X^{j})_i[/itex].

If [itex]k=2[/itex] and [itex]n=3[/itex], then the solution is [itex]\|X^1\times X^2\|[/itex].

I know that a wedge product exists between alternating multilinear forms, and that it is related to measures because it is used in differential geometry and integration, but the definition of the wedge product doesn't immediately answer my question.

If [itex]k=n[/itex], then the solution is [itex]|\textrm{det}(X)|[/itex] with [itex]X_{ij}=(X^{j})_i[/itex].

If [itex]k=2[/itex] and [itex]n=3[/itex], then the solution is [itex]\|X^1\times X^2\|[/itex].

I know that a wedge product exists between alternating multilinear forms, and that it is related to measures because it is used in differential geometry and integration, but the definition of the wedge product doesn't immediately answer my question.