# Basis, set of matrices, proof...

• May 26th 2011, 08:33 PM
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 \$\displaystyle M_{22}\$, do I need to show that the set spans \$\displaystyle M_{22}\$, and is linearly independent? Or can I just show that they are linearly independent to prove that they are a basis for \$\displaystyle M_{22}\$??? Don't I know that \$\displaystyle 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 26th 2011, 11:12 PM
FernandoRevilla
Quote:

...\$\displaystyle 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, 09:06 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.