Let $(u_1, \ldots, u_n)$ and $(v_1, \ldots, v_n)$ be two ordered bases of $\mathbb R^n$. The orientation of the first basis is defined as the sign of the determinant of $[u_1 \cdots u_n]$, and similarly for the second basis. Prove that the first basis can be continuously transformed into the second one, while remaining linearly independent at all times, if and only if the two bases have the same orientation.

The "only if" direction is easy, because the determinant, which must change continuously, cannot change from positive to negative without going through zero. I'm looking for a proof of the "if" part.

More broadly, I'm looking for comments on the issue of defining the determinant in a nice way. The definitions I've seen say something like:

"The determinant is a quantity that has some nice properties. For one, the determinant is zero if and only if the corresponding matrix is singular. Furthermore, its absolute value equals the volume of the parallelepiped spanned by the vectors. And the sign corresponds to the orientation of the vectors. And what is the "orientation" of a tuple of vectors? Well, it's defined as the sign of the determinant!"

The above claim, if correct, might lead to a more natural (and less circular) definition of orientation, and also of the determinant.

Also, is it necessary to *define* the n-dimensional volume of a parallelepiped as the absolute value of the determinant (as I have seen in some places)? Can't they be *shown* to be equal via elementary arguments? Consider the "cut-and-paste" proof that the area of a parallelogram equals the area of a rectangle with the same base and height. I think a similar n-dimensional cut-and-paste can show that

$$\mathrm{vol}(u_1, u_2, \cdots , u_n) = \mathrm{vol}(u_1 + k u_2, u_2, \cdots, u_n),$$

and similarly for the other elementary properties of the determinant. But I haven't thought it through.

Thanks in advance!

preciselytwo components, and not more? – Theo Johnson-Freyd Nov 11 '12 at 22:07teachthe definition of a determinant, and isn't completely satisfied with the approaches seen in the books I've consulted for teaching: I vote to re-open. It might not be "research mathematics" but it is something where a research-level perspective might illuminate the pedagogical niceties. – Yemon Choi Nov 11 '12 at 22:12