Linear Algebra Done Right · Problem 3E.18

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · quotient-spaces

Problem

Suppose that

U

is a subspace of

V

such that

V/U

is finite-dimensional. (a) Show that if

W

is a finite-dimensional subspace of

V

and

V = U + W

, then

\dim W \geq \dim V/U

. (b) Prove that there exists a finite-dimensional subspace

W

V

such that

\dim W = \dim V/U

and

V = U \oplus W