An upper bound on the number of nonzero eigenvalues

#linear_algebra

Suppose $V$ is a finite-dimensional real vector space and $T : V \to V$ is a linear transformation. Prove that $T$ has at most $\dim (range T)$ distinct nonzero eigenvalues.

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Suppose $V$ is a finite-dimensional real vector space and $T:V\to V$ is a linear transformation. Prove that $T$ has at most $\dim(\operatorname{range} \,T)$ distinct nonzero eigenvalues.