Endomorphism/Eigenvectors/Linearly independent/Fact/Proof

Proof

We prove the statement by induction on . For , the statement is true. Suppose now that the statement is true for less than vectors. We consider a representation of , say

We apply to this and get, on one hand,

On the other hand, we multiply the equation with and get

We look at the difference of the two equations, and get

By the induction hypothesis, we get for the coefficients , . Because of , we get  for , and because of , we also get .