Linear Algebra Done Right · Problem 5D.4

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · diagonalizable-operators

Problem

Suppose

V

is finite-dimensional and

T \in \mathcal{L}(V)

. Prove that the following are equivalent. ((a))

V = \operatorname{null} T \oplus \operatorname{range} T

. ((b))

V = \operatorname{null} T + \operatorname{range} T

. ((c))

\operatorname{null} T \cap \operatorname{range} T = \{0\}