Hi, I was just wondering if there is something more to the concept of an ordered basis other than the fact that it is simply a basis which is ordered. The reason I'm asking this is because I don't know why some linear algebra books consider this important enough to make the distinction. I mean, given a basis for a finite-dimensional vector space, we can always order it any way we choose. In fact, whenever we talk about a finite set in general, we automatically give it an ordering so we are able to talk about it meaningfully.
It is just to avoid some confusion. It is used both for identification and for uniqueness. Say we have an ordered basis {v1,v2,v3}. We say each vector v may be written v=a1v1+a2v2+a3v3. Now in speaking of v we avoid constantly making comments like with "no to basis vectors equal" and "with the basis vectors ordered as before".
So, in lurflurf's example we could write v as (a1, a2, a3). Using the same basis vectors but with a different order would interchange those numbers. Also, a linear transformation, from one vector space to another, can be written as matrix given a basis for each space. Of course, the matrix depends upon the basis and upon the order. Using exactly the same bases but changing the order would interchange the numbers in the matrix representation. To use those representations, vectors as ordered n-tuples and matrices we must specify not just the basis but the order of the basis vectors.
Thanks. That clears it up for me. So, I guess specifying a finite basis without an ordering is not very useful. I mean, I don't even see a way to talk about any finite set without first indexing the elements and assuming the natural ordering.