lect136_15_w09

# lect136_15_w09 - Wednesday February 6 Lecture 15 More on...

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Wednesday February 6 Lecture 15: More on abstract vectors spaces and subspaces (Refers to section 4.1) Expectations: 1. Determine if a given set is a vector space or not. 2. Recognize linear transformations from one vector space to another. 15.1 Definition –Let { v 1 , v 2 , …, v m } be a finite set of vectors in a vector space V . If every vector v in V is a linear combination of v 1 , v 2 , …, v m then we say that spans i and write span{ v 1 , v 2 , …, v m }. 15.1.1 Remark – As for R n if W = span{ u 1 , u 2 , …, u m } inside a vector space V the W is a subspace of V . 15.1.2 Example – We show in class that the set {1, x , x 2 , …, x n } spans P n , i.e., P n = span{1, x , x 2 , …, x n }. 15.2 Definition – A function T which maps vectors in a vector space V 1 to vectors of another vector space V 2 is called a linear map if it satisfies the condition T ( α v + β w ) = α T ( v ) + β T ( w ). Note that the vectors spaces need not be similar in nature. That is

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lect136_15_w09 - Wednesday February 6 Lecture 15 More on...

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