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## Homework Statement

Let V be a finite-dimensional vector space and let T: V -> V be a linear transformation. Prove that there exists a natural number n so that

Ker(T^n) (intersection) Range(T^n) = {0}

Here, T^n represents the n-fold composition of T o T o ... o T

## Homework Equations

## The Attempt at a Solution

I can prove that Ker(T) (intersection) Range(T) = {0} by showing that if an element is in the intersection of the two spaces, it must be zero since would produce a linear combination of one basis equal to the other, and subtracting so they are on both sides gives a linear combination of both bases. But the sum of the spans of the two bases are linearly independent, so the coefficients must all be zero, and thus they can have only the zero vector in common. Now I'm unsure of how to relate this to T^n.