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**Homework Statement**

Prove that if there exists a linear map on V whose null space and range are both finite dimensional, then V is finite dimensional.

**The attempt at a solution**

I *think* the following is true: For all v in V, T(v) is in range(T), otherwise T(v) = 0 which implies v is in null (T).

Other than that, I know I can write a basis {v_1, ..., v_n} for null(T) and a basis {T(u_1), ..., T(u_m)} for range(T), where range(T) = {T(u) : u is in V}. But since this is a linear map {u_1, ..., u_m} should also be a basis for some U such that U is a subspace of V.

Does anyone know if these assumptions are heading in the right direction?