Basis, set of matrices, proof...

If I have a set of 2X2 matrices A1, A2, A3, A4 and I want to show that they form a basis for the set of all 2X2 matrices , do I need to show that the set spans , and is linearly independent? Or can I just show that they are linearly independent to prove that they are a basis for ??? Don't I know that is a four dimensional vector space, and since I have a set of 4 matrices, I only need to show spanning or linear independence right?

(Thinking)

Quote:

---

Originally Posted by adkinsjr

... is a four dimensional vector space, and since I have a set of 4 matrices, I only need to show spanning or linear independence right?

---

Right.

A basis for any n dimensional vector space (finite dimensional, of course) has 3 properties
1) They span the space
2) They are independent
3) There are n vectors in the basis

And if any two of those are true, the third is also.