Finding the Dimension and Basis of the Matrix Vector space

[nautolian]

Homework Statement

The set K of 2 × 2 real matrices of the form [a b, -b a] form a field with the usual operations.
It should be clear to you that M₂₂(R) is a vector space over K. What is the dimension of M₂₂(R) over K? Justify your answer by displaying a basis and proving that the set displayed is actually a basis.

Homework Equations

The Attempt at a Solution

I don't think there can be a basis over the field K, because no linear combination of the matrices [a b, -b a] with any M₂₂ can form, say [1 0, 0 0]. Which would be in M₂₂. Any help would be greatly appreciated. Thanks!

 

[nautolian]

Could you do a combination say (0 a, 0 -b) (0 -b, 0 a), (a 0, b 0), (-b 0, a 0)? Could this form a basis? Or does it have to be real integers? thanks