Show 3. Showing Linear Dependence that the set is linearly dependent by finding a nontriviai linear combination of vectors in the set whose sum is the zero vector. Then express one of the vectors in the set as a linear combination of the other vectors in the set. a).s = {(3, 4), (–1, 1), (2, 0)} b). s = {(1, 1, 1), (1, 1, 0), (0, 1, 1), (0, 0, 1)}

Show 3. Showing Linear Dependence that the set is linearly dependent by finding a nontriviai linear combination of vectors in the set whose sum is the zero vector. Then express one of the vectors in the set as a linear combination of the other vectors in the set. a).s = {(3, 4), (–1, 1), (2, 0)} b). s = {(1, 1, 1), (1, 1, 0), (0, 1, 1), (0, 0, 1)}

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.4: Spanning Sets And Linear Independence

Problem 70E: Proof When the set of vectors {u1,u2,...,un} is linearly independent and the set {u1,u2,...,un,v} is...

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Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. a Explain what it means to say that a set of vectors is linearly independent. b Determine whether the set S is linearly dependent or independent. S={(1,0,1,0),(0,3,0,1),(1,1,2,2),(3,4,1,2)}
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Proof When the set of vectors {u1,u2,...,un} is linearly independent and the set {u1,u2,...,un,v} is linearly dependent, prove that v is the linear combination of the uis.
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Guided Proof 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 c1v1+c2v2+...+ckvk=0. (iii) Use this fact to derive a contradiction and conclude that T is linearly independent.
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Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set {v12v2,2v23v3,3v3v1} linearly dependent or linearly independent? Explain.
Proof Prove that if S1 and S2 are orthogonal subspaces of Rn, then their intersection consists of only the zero vector.
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Proof Let S={u,v} be a linearly independent set. Prove that the set {u+v,uv} is linearly independent.