- #1

Bipolarity

- 775

- 2

Let V be a vector space of dimension n and let S be a generating set for V. Prove that some subset of S is a basis for V.

The argument is as follows:

If ##V = \{ 0 \} ## then it is trivial since the null set is a basis for V. Otherwise V has dimension greater than zero, so that S is nonempty.

Given that S is nonempty, take a new set S' which is empty. Look at all the vectors in S and chuck them into S' making sure that independence of S' is preserved. A point will come where S has n vectors, at which point it is a basis.

The problem is the termination of this process and the procedure by which vectors in S are taken and chucked into S'. If S is finite, all is well, since we can search through all the elements of S in a finite amount of time. But what if S is infinite?

BiP