Let $U$ be a finite-dimensional real vector space and $b$ a positive definite symmetric bilinear form on $U \times U$. Let $\psi: U \rightarrow U$ be a linear map such that $b(\psi(x), y)+b(x, \psi(y))=0$ for all $x$ and $y$ in $U$. Prove that if $\psi$ is invertible, then the dimension of $U$ must be even. By considering the restriction of $\psi$ to its image or otherwise, prove that the rank of $\psi$ is always even.