The question reads:
Let be vectors in , and assume that . Prove that there is a symmetric matrix such that .
I don't really know where to begin on this. It's in the chapter on bilinear forms, but I don't see what this has to do with that. Maybe find some vector such that , so that we have to solve , in other words, show that there exists a vector that is orthogonal to with respect to the bilinear form (and then maybe use the fact that the eigenvectors of form an orthogonal basis).
But other than that speculation, I got nothing.
Help is appreciated!