Number of Elements in Basis Sets for V: Prove It!

[FourierX]

Homework Statement

Prove:

Any two basis sets for V have the same number of elements.

Homework Equations

The Attempt at a Solution

Sounds obvious but is quite intricate to prove it.

 

[Dick]

Is it? I don't agree, at least if the dimension V is finite.

 

[FourierX]

You confused me even more :(

 

[HallsofIvy]

He said he did not agree that the proof is quite intricate.

I know, it always confuses me when people don't agree with me, too.

A space is said to be finite dimensional if and only if there exist a finite spanning set. In that case, since the number of vectors in a spanning set is an integer, there must exist a smallest spanning set. Since a basis is a set of vectors that is both a spanning set and independent you need to prove:

1) The smallest spanning set is independent. (Show that if it were not a independent, you could remove one of the vectors and still have a spanning set, contradicting the fact that it is smallest.)

2) No set of independent vectors can have more members than the smallest spanning set. (Take a supposedly independent set with more vectors and rewrite each in terms of the smallest spanning set.)