# Basis, set of matrices, proof...

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• May 26th 2011, 09:33 PM
adkinsjr
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 $M_{22}$, do I need to show that the set spans $M_{22}$, and is linearly independent? Or can I just show that they are linearly independent to prove that they are a basis for $M_{22}$??? Don't I know that $M_{22}$ 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)
• May 27th 2011, 12:12 AM
FernandoRevilla
Quote:

Originally Posted by adkinsjr
... $M_{22}$ 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.
• May 27th 2011, 10:06 AM
HallsofIvy
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.