Linear Algebra Done Right · Problem 3E.12

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · quotient-spaces

Problem

Suppose

v_1, \ldots, v_m \in V

. Let

A = \{\lambda_1 v_1 + \cdots + \lambda_m v_m : \lambda_1, \ldots, \lambda_m \in \mathbf{F} \text{ and } \lambda_1 + \cdots + \lambda_m = 1\}.

(a) Prove that

A

is a translate of some subspace of

V

. (b) Prove that if

B

is a translate of some subspace of

V

and

\{v_1, \ldots, v_m\} \subseteq B

, then

A \subseteq B

. (c) Prove that

A

is a translate of some subspace of

V

of dimension less than

m