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.
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!