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?