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!