Question

Give any two bases for M22(R), the set of 2x2 matrices.

I know that the following is a basis, but I don't know how to make another one:

1 0 0 1 0 0 0 0
0 0 0 0 1 0 0 1

Answer 1

e1,e2,e3,e4 are the above basis,any linear combination of these basis ae1+be2,ce3,de4,e2,e3,e4 will also form a basis.ae2,be2,ce3,de4 will also form a basis.we can construct any linear combination of vectors using these basis vector,as long as the new vectors are linearly independent we can consider it as new basis of vector space