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 I be a subset of S. (ii) If T is linearly dependent, then there exist constants not all zero satisfying the vector equation c, v, + c,v, + + CV = 0. (iii) Use this fact to derive a contradiction and conclude that T is linearly independent. O v,eS and S is linearly independent O v, ES and S is linearly dependent O v,ES and S is linearly independent O v,ES 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 is a set of linearly independent vectors. Let I be a subset of S. (ii) If T is linearly dependent, then there exist constants not all zero satisfying the vector equation c, v, + c,v, + + CV = 0. (iii) Use this fact to derive a contradiction and conclude that T is linearly independent. O v,eS and S is linearly independent O v, ES and S is linearly dependent O v,ES and S is linearly independent O v,ES 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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