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
Related questions
Question
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning