Basis, linearly independence, n elements, proof

Question

Let V be a vector space of dimension n and consider any finite subset S ⊂ V.

I want to try to formulate a proof for the following statement: "if S has n elements and is linearly independent, then it is a basis"

should I prove that if S has n elements and is linearly independent then it spans V therefore it is a basis.

Similarly, if the statement was "if S has n elements and spans V, then it is a basis" should I show that if S has n elements and spans V then it is linearly independent so it is a basis? if yes, how do I show this? If not, where am I wrong? what's the correct way?

Answer 1

These two results are consequences of the following three facts:

• Every spanning list can be reduced to a basis.
• Every linearly independent list (in a finite-dimensional space) may be extended to a basis.
• Every (finite) basis has the same length.

If you have a spanning list with $n$ vectors, then you may extend it to a basis, necessarily containing $n$ vectors. This means you added nothing to it, hence the original $n$ vectors were a basis. Similar reasoning works for the linearly independent case too.

As for proving the above results, it requires a little bit of fiddling around. It's nothing too hard, but it's long and tedious. There's usually some lemma that needs to be proven; I've seen such lemmas called the Exchange Lemma and/or the Linear Dependence Lemma.