Thread: Basis

Exhibit a basis for the given space and prove that it is a basis.

The space of 2 x 2 matrices.

I don't understand how to this at all, can someone take me through this example step by step and kind of explain why/concepts involved.

Thanks so much.

Originally Posted by millerst

Exhibit a basis for the given space and prove that it is a basis.

The space of 2 x 2 matrices.

I don't understand how to this at all, can someone take me through this example step by step and kind of explain why/concepts involved.

Thanks so much.

It's the 2x2 identity matrix, since it is linearly independent and spanning.

I can't understand what Roam said at all since the words "independent" and "spanning" apply to sets of vectors, not individual vectors or matrices.

millerst, you probably know that a basis for, say, , the set of quadruples, (a, b, c, d), is {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)} since we can write (a, b, c, d)= (a, 0, 0, 0)+ (0, b, 0, 0)+ (0, 0, c, 0)+ (0, 0, 0, d)= a(1, 0, 0, 0)+ b(0, 1, 0, 0)+ c(0, 0, 1, 0)+ d(0, 0, 0, 1).

Okay, any matrix in your space can be written as


You want matrices M1, M2, M3, M4 so that

there are obvious matrices that do that.

The standard basis for consists of the following four matrices:



Because the space of 2x2 matrices has dimension 4, so can't be spanned by just two vectors.

Sorry for my carelessness...