# Proving a property of the dimension of eigenspaces in a finite dimensional space

cloverforce

## Homework Statement

Prove that if A: V - >V is a linear map, dim V = n, and h1,...,hk (where 1,...,k are subscripts) are pairwise different eigenvalues of A such that their geometric multiplicities sum to n, then A does not have any other eigenvalues.

## Homework Equations

Note sure if this is relevant or not, but if hj and hk are distinct eigenvalues, then the intersection of V(hj) and V(hk) is {0}.

## The Attempt at a Solution

My attempt has been by contradiction. Suppose that there exists an eigenvalue h distinct from the k terms already given. In that case, I want to show that dim(V(h)) = 0, which would mean that its basis is the empty set and thus no such eigenvalue can exist. I'm not sure why this would have to be the case, though.

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