Proving Linear Independence and Span of D

[andytoh]

I can't seem to figure this one out:

Question: Let D be a nonempty subset of a vector space V over a field F. Let B be a finite linearly independet subset of span D having n elements. Prove there exists a subset D' of D also having n elements such that

span[(D-D') U B] = span(D).

Moreover, if D is linearly independent, so is (D-D') U B.

Can anyone help?

 

[andytoh]

Here's the beginning of my induction proof for the case of 1 and 2 elements. I'm working on the nth step now.

 

[andytoh]

Ok, I've finished the proof now.

It is about 2 pages long! Even the base case for 1 element had to be modified to some length.