Linear Independence and Dependency in Vector Spaces
Explore the concepts of linear independence and dependency in vector spaces through a series of proofs and examples. Understand the relationships between distinct vectors, linearly independent sets, and linear transformations within the context of linear algebra.
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Presentation Transcript
Let ?1,?2,,?? be a linearly independent set of vectors in ??, and let ?be a vector in ??such that ? = ?1?1+ ?2?2+ + ???? for some scalars ?1,?2, ,?? with ?1 0. Prove that ?,?2, ,?? is linear independent. 86
Let ? and ? be distinct vectors in ??. Prove that the set ?,? is linearly independent if and only if the set ? + ?,? ? is linearly independent. 87
Prove that if ?1,?2,,?? is a linearly independent subset of ?? and ?1,?2, ,?? are nonzero scalars, then ?1?1,?2?2, ,???? is also linearly independent. 89
Let S = ?1,?2,,?? be a nonempty set of vectors from ??. Prove that if ? is linearly independent, then every vector in ???? ? can be written as ?1?1+ ?2?2+ + ???? for unique scalars ?1,?2, ,??. 93
Let S = ?1,?2,,?? be a nonempty set of vectors from ??. Prove that if ? is linearly independent, then every vector in ???? ? can be written as ?1?1+ ?2?2+ + ???? for unique scalars ?1,?2, ,??. State and prove the converse of Exercise 93. 94
Let S = ?1,?2,,?? be a nonempty subset of ??and ? be an ? ? matrix. Prove that if ? is linearly dependent, and ? = ??1,??2, ,??? contains ? distinct vectors, then ? is linearly dependent. 95
Let S = ?1,?2,,?? be a nonempty subset of ??and ? be an ? ? matrix. Prove that if ? is linearly dependent, and ? = ??1,??2, ,??? contains ? distinct vectors, then ? is linearly dependent. 96 Give an example to show that the preceding exercise is false if linearly dependent is changed to linearly independent.
