Guided Proof Prove that a nonempty subset of afinite set of linearly independent vectors

Chapter 4, Problem 67

(choose chapter or problem)

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 \(c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}+\cdots+c_{k} \mathbf{v}_{k}=\mathbf{0}\).

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

Text Transcription:

c_1 mathbf v_1 +c_2 mathbf v_2 +cdots+c_k mathbf v_k =mathbf 0