Let $\phi: G \rightarrow GL(V)$ be a representation with character $\chi$. Let $W$ be the subspace $\{v \in V : \phi(g).v = v$ for all $g \in G \}$ of $V$. Prove that $\dim W = \langle \chi, \chi_\text{Trivial}\rangle$.
I've thought that the character is the trace, therefore the sum of the eigenvalues. And, in this case, the vectors of $W$ have eigenvalue $1$. I'm not sure how to relate this back to the dimension of $W$ though... Any suggestions?
The notation $\chi_\text{Trivial}$ is very nonstandard - I assume you mean it to refer to the trivial character?
I'm also going to assume that we're talking about finite groups and complex representations. Think about the decomposition of $\chi_V$ into irreducible characters, and what the multiplicity will be for the trivial character.