Let M_2x2 denote the space of 2x2 matrices with real coeffcients. Show that (a1 b1) . (a2 b2) (c1 d1) (c2 d2) = a1a2 + 2b1b2 + c1c2 + 2d1d2 defines an inner product on M_2x2. Find an orthogonal basis of the subspace S = (a b) such that a + 3b - c = 0 (c d) of M_2x2 defined by with respect to this inner product. I know how to find the orthogonal basis, so I don't think I need any help with that, I'm only having trouble with the first part--showing that the two matrices define inner product. I have no idea where the 2's in 2b1b2 and 2d1d2 are coming from. I thought that it should be just a1a2 + b1b2 + c1c2 + d1d2, without the 2's.