Hi, All: I am trying to see how to classify all symmetric bilinear forms B on R^3 as a V.Space over the reals. My idea is to use the standard basis for R^3 , then use the matrix representation M =x^T.M.y . Then, since M is, by assumption, symmetric, we can diagonalize M. So it seems all symmetric bilinear forms are just all diagonal matrices; maybe we need to factor out those that are equivalent as bilinear forms, i.e., B,B' are equivalent if there is a linear isomorphism L:R^3-->R^3 with B(v)=B'(L(v)). Is this right? Is there anything else?