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 i a set of linearly independent vectors. Let T be a subset of S. (ii) If Tis linearly dependent, then there exist constants not all zero satisfying the vector equation c,v, + c2v2 + ... + C&V¢ = 0. (iii) Use this fact to derive a contradiction and conclude that Tis linearly independent. O v, ES and S is linearly dependent v, ¢S and S is linearly independent v, ES and S is linearly independent v, ¢s and S is linearly dependent So, Tis linearly independent.

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 i a set of linearly independent vectors. Let T be a subset of S. (ii) If Tis linearly dependent, then there exist constants not all zero satisfying the vector equation c,v, + c2v2 + ... + C&V¢ = 0. (iii) Use this fact to derive a contradiction and conclude that Tis linearly independent. O v, ES and S is linearly dependent v, ¢S and S is linearly independent v, ES and S is linearly independent v, ¢s and S is linearly dependent So, Tis linearly independent.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.3: Spanning Sets And Linear Independence

Problem 30EQ

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