No linear relation between matrix coefficients of all the irreducible repn of finite group

Fix a finite group $G$, and look at all its irreducible representations/$\mathbb{C}$. It is said in Serre's book that "there cannot be any $\mathbb{C}$-linear relation between the matrix coefficients of these irreducible representations (of course we fix a set of bases first), because of the orthogonality formulas". I don't understand the quoted part. Can anyone explain that? I think I must be missing something obvious here.

-

Here orthogonality is with respect to a complex (sesquilinear) positive-definite inner product, and we can regard each matrix entry as a vector of length $|G|$.