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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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