Continuous change of basis (and on the definition of determinant) 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!
 A: Basically what you're asking in the first question is whether $SL_n({\mathbb R})$ is pathwise connected.  Using the polar decomposition, this follows from the fact that $SO_n({\mathbb R})$ is pathwise connected.  That, in turn, follows from the properties of Givens rotations.
A: About volume: first, I think it is reasonable, especially when addressing beginning students, to admit that we do have a prior notion of "volume", even if it is imprecise. Probably most people would agree that rigid rotations (=$K=SO(n,\mathbb R)$) preserve volume, and that dilation (by positive amounts) of coordinates (=$A^+$, the positive diagonal matrices) multiply by the product of the diagonal entries, by looking at boxes with sides parallels to the axes. Then by a Cartan decomposition (or some more elementary-sounding name, if desired) $K\cdot A^+ \cdot K$ is close to being $GL(n,\mathbb R)$: it is all positive-determinant matrices. Thus, we've given an "accommodating" argument that all matrices with positive determinant change volume by the determinant.
Robert Israel's point about path-connectedness can be re-used in this context.
A: Re your general questions: in multilinear algebra, the determinant (of $n$-tuples of vectors with $n$ coordinates in some previously chosen ordered basis) is often defined as the (unique) $n$-linear alternating form which takes value $1$ on the basis.  From this the properties of determinant follow almost tautologically: the alternating property causes determinant to be zero on linearly dependent $n$-tuples, and changing the basis requires scaling the determinant by a factor. Also, the definition of the volume of a parallelepiped becomes natural.
Using this definition of determinant, the orientation can be defined as follows (cutting and pasting from Wikipedia): ``Let $V$ be a finite-dimensional real vector space and let $b_1$ and $b_2$ be two ordered bases for $V$. It is a standard result in linear algebra that there exists a unique linear transformation $A : V \to V$ that takes $b_1$ to $b_2$. The bases $b_1$ and $b_2$ are said to have the same orientation (or be consistently oriented) if $A$ has positive determinant; otherwise they have opposite orientations. The property of having the same orientation defines an equivalence relation on the set of all ordered bases for $V$. If $V$ is non-zero, there are precisely two equivalence classes determined by this relation. An orientation on $V$ is an assignment of $+1$ to one equivalence class and $−1$ to the other." 
