Let be a field, a vector space over , and a linear transformation. Let be a finite set of eigenvalues for , and let be a nonzero eigenvector for for each . (That is, .) Prove that the are linearly independent. Conclude that any linear transformation on a vector space of finite dimension has at most eigenvalues.
We proceed by induction on .
For the base case , certainly is linearly independent since is torsion-free.
Suppose now that any set of eigenvectors for distinct eigenvalues is linearly independent, and let be a set of eigenvectors for the distinct eigenvalues . Suppose we have such that . Now . Suppose, without loss of generality, that . Now , so that . Now , and by our hypothesis, for all . But then we have , a contradiction since and . Thus in the sum , all of the are zero. Hence is linearly independent.
The dimension of a vector space is the maximal cardinality of a linearly independent subset. In particular, we see that if has finite dimension , then can have at most eigenvalues.