A basis to a orthogonal set Let $(E, \langle \cdot, \cdot \rangle)$ be an $n$-dimensional Hilbert space and $A,B \colon E \to E$ linear isomorphisms. 
Does there exist a basis $\{e_{1},...,e_{n}\}$ of $E$ such that $\mathcal{B}=\{A(e_{1}),...,A(e_{n}),B(e_{1}),...,B(e_{n})\}$  are a orthogonal set?
Hints or solutions are greatly appreciated.
Related: Find a basis for two operators

Nope, this is not possible, since if $e_i$ form a basis of E, any other vector $v \in E$ is also satisfies $v \in span(\{e_i\})$, so any vector in B can be expressed as a linear combination of $\{e_i\}$.
