Hiya,

Hate to ask a question without essentially any work on it, but I have no idea how to prove this. The question is : Let $F$ be the matrix of a nonsingular bilinear form $f$ on a real space of dimension n. Prove that for odd $n$ the matrix $−F$ is not the matrix of $f$ in any basis of $V$.

I'm just not sure how to tackle this, on the one hand it seems obvious that $f(x,y) = x^TFy \not = x^T(-F)Y$, assuming $F \not =0$ (which it isn't since it's non-singular), but then I suppose there's something about different bases making this not so trivial as saying "a thing is not equal to its negative".

I apologise again but I really haven't got any idea. Any help would be greatly appreciated, thank you very much in advance.