1. The problem statement, all variables and given/known data T is a transformation from the vector space of real 2x2 matrices back to that space. T(X) = X - trans(X) (trans = transposed) a)Find a base for KerT and ImT b)Prove that T can be diagonalized. 2. Relevant equations 3. The attempt at a solution a) If X is in KerT then X = trans(X) and so KerT is the space of all 2x2 symmetric matrices and its dimension is 3. It's base is any base of that space. Since dimKerT = 3 , dimImT = 1 and since [0 1] [-1 0] is in the image, it is a base of it. b) obviously 0 is an eigenvalue with three linearly independant eigenvectors. It's easy to see that 2 is an eigenvalue for all the vectors in the image and so we have four linearly independent eigenvectors which means that T can be diagonalized. Is that right? (b) seems a little shaky to me, is there any way i can say it better? Thanks.