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Lecture3 - Lecture 3 3.1 Linear dependence Let X = cfw v1 v...

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1 Lecture 3 3.1 Linear dependence Let X = { 1 v , 2 v , …, k v } be a set of vectors in a vector space V. If r 1 1 v + r 2 2 v + … + r k k v = 0 for some 0 j r , j = 1, 2, …, k, then X is a linearly dependent set. Otherwise, X is a linearly independent set. Example: Is the set {sin 2 x, cos 2 x,1} linearly dependent or independent? Examples: In each case determine if the set of vectors is independence. { 0 1 2 1 , 3 1 0 2 , 6 5 6 1 }, {e x , e 2x }, {1, x, x 2 , …, x n , …} Example: Let V denote a vector space. Prove: 1. If 0 v in V, then { v } is an independent set. 2. No independent set of vectors in V can contain the zero vector. 3. If { 1 v , 2 v , …, k v } is independent in V, so is { a 1 1 v , a 2 2 v , …, a k k v } if 0 i a .

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2 Example: Suppose that A is an n n matrix such that A n = 0 but A n 1 0. Show that B = {I, A, A 2 , …, A n 1 } is independent in M nn .
3 3.2 Bases A set of vectors in a vector space V is a basis of V if 1. The set of vectors span V 2. The set of vectors is linearly independent.

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