Orthogonal complement to a graph of a linear map

Question

Let $E$ and $F$ be two Euclidean/Hermitian vector spaces and $f:E\rightarrow F$ a linear map. Let $\mathcal{G}(f)<E\oplus F$ be the graph of $f$.

Assume if it helps that $\{e_i\}_i$ is an orthonormal basis of $E$, so that $\{(e_i,f(e_i)\}_i$ is a basis of $\mathcal{G}(f)$.

Is there a simple way to describe:

1- A basis of the orthogonal complement of $\mathcal{G}(f)$ in $E\oplus F$ (where we put that $E$ and $F$ are orthogonal) ?

2- A basis of any complement fo $\mathcal{G}(f)$?

Thanks!

Answer 1

There is a simple way indeed. The orthogonal complement $\mathcal{G}(f)^\perp$ is given by the graph $\mathcal{G}(-f^*)$, where $f^*\colon F\to E$ is the (Hermitian) adjoint of $f\colon E\to F$. The proof is simply: $$ \langle (u,f(u)), (-f^*(v),v)\rangle_{E\oplus F} = \langle u,-f^*(v)\rangle_E+\langle f(u),v\rangle_F=0 $$ for all $u\in E$ and all $v\in F$. So you can just pick any basis of $F$ and write down a basis of $\mathcal{G}(-f^*)$.