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**Unformatted text preview: **Math 205, Spring 2011 Homework 6: due Feb 28 (Mon) Chapter 4, Sections 3, 4 and 5. Week 6: 4.3 Subspaces/Nullspace 4.4 Spanning Sets 4.5 Linear Dependence, Linear Independence ———– Recall that a Vector Space V may be any collection with formulas for vector plus and scalar mult, V = ( V , + , · ) with 10 rules. In practice, in Math 205, Vector Spaces will usually be subsets of one of four standard Examples: R n ; V = M m × n ( R ); C k ( a, b ) = functions: f+g, k · f, 0, -f with k derivatives P n = polynomials with real coef. degree < n. 2 We usually assume the 10 rules for these four vector spaces (the rules are easier to verify than to remember). So to establish that a subset S of one of these spaces is a vector space we only need to check the last two rules, in which case we say that S is a Subspace of V. The two subspace conditions are the closure rules (1) for every vectoru,vectorv ∈ S, check vectoru + vectorv ∈ S (closure under vector +), and (2) for every vectoru ∈ S, and every scalar c ∈ R , check cvectoru ∈ S (closure under scalar mult). For the formal definition of span and spanning set we take vectors vectorv 1 ,vectorv 2 , . . . ,vectorv k in a vector space V. For the most part,...

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