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cloverforce
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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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