Abstract Algebra: Theory and Applications · Problem 20.12

Fields — Thomas Judson

Mathematics — Abstract Algebra · linear-independence

Problem

Prove that any set of vectors containing

\mathbf{0}

is linearly dependent. Hint. Let

v_0 = \mathbf{0}, v_1, \ldots, v_n \in V

and

\alpha_0 \neq 0, \alpha_1, \ldots, \alpha_n \in F

. Then

\alpha_0 v_0 + \cdots + \alpha_n v_n = \mathbf{0}