Chapter 4.4, Problem 67E

Guided Proof Prove that a nonempty subset of a finite set of linearly independent vectors is linearly independent.

Getting Started: You need to show that a subset of a linearly independent set of vectors cannot be linearly dependent.

(i) Assume $S$ is a set of linearly independent vectors. Let $T$ be a subset of $S$.

(ii) If $T$ is linearly dependent, then there exist constants not all zero satisfying the vector equation $c_1{v}_1+c_2{v}_2+\mathrm{...}+c_k{v}_k=0$.

(iii) Use this fact to derive a contradiction and conclude that $T$ is linearly independent.