You have the set of all 2 by 2 matrices of the form and you want to
1) Show that this is a subspace of the space of all 2 by 2 matrices.
You must show it is closed under matrix addition and scalar multiplication. The sum of two such matrices can be written
Can that be written in the form ? What must a, b, and d be?
The product of the number "k" and the matrix is .
Can that be written in the form ? What must a, b, and d be?
The third condition, that the subset be non-empty, can be done by showing that the 0 matrix is in the set. Take a= b= d= 0.
2) Find a basis for this subspace.
Well, . Does that give you any ideas?
3) Find the dimension.
After (2), this should be trivial.