Linear Algebra Done Right · Problem 3E.7

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · quotient-spaces

Problem

Let

U = \{(x, y, z) \in \mathbf{R}^3 : 2x + 3y + 5z = 0\}

. Suppose

A \subseteq \mathbf{R}^3

. Prove that

A

is a translate of

U

if and only if there exists

c \in \mathbf{R}

such that

A = \{(x, y, z) \in \mathbf{R}^3 : 2x + 3y + 5z = c\}.