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. (1) 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,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 dependent O v,ES and S is linearly independent O v, ¢S and S is linearly dependent O v, ¢S and S is linearly independent So, Tis linearly independent. Need Help? Read It
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. (1) 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,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 dependent O v,ES and S is linearly independent O v, ¢S and S is linearly dependent O v, ¢S and S is linearly independent So, Tis linearly independent. Need Help? Read It
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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![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, v, + c,v, + …*
+
= 0.
(iii) Use this fact to derive a contradiction and conclude that T is linearly independent.
v, ES and S is linearly dependent
v, ES and S is linearly independent
v, ¢S and S is linearly dependent
v, ¢S and S is linearly independent
V
So, T is linearly independent.
Need Help?
Read It](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e6ac768-d400-4f51-974f-f918e07c8ccd%2F3323c9fd-92c9-4a5d-b50e-d1e76be59103%2Fmhjgmts_processed.png&w=3840&q=75)
Transcribed Image Text: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, v, + c,v, + …*
+
= 0.
(iii) Use this fact to derive a contradiction and conclude that T is linearly independent.
v, ES and S is linearly dependent
v, ES and S is linearly independent
v, ¢S and S is linearly dependent
v, ¢S and S is linearly independent
V
So, T is linearly independent.
Need Help?
Read It
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