Kunze Linear Algebra · Problem 3.2.11

Linear Transformations

Mathematics — Methods · range-null-disjoint-rank-condition

Problem

Let

V

be a finite-dimensional vector space and let

T

be a linear operator on

V

. Suppose that

\text{rank}(T^2) = \text{rank}(T)

. Prove that the range and null space of

T

are disjoint, i.e., have only the zero vector in common.