Given V_R = R^2, then let V_R have the standard basis B=B'={(1,0), (0,1)} and let L_theta ((1,0)) = (cos theta, sin theta), L_theta ((0,1)) = (-sin theta, cos theta).

I was told that this matrix is the standard rotation matrix. Now I have a question that says:

Determine the matrix associated (relative to the basis B) with the linear transformation L_theta.

Since the basis B is the standard basis, need I do anything to the 2x2 rotation matrix in order to present it "in terms of" this basis? If so, what do I need to do?