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Unformatted text preview: HOMEWORK #3 DUE 5pm WEDNESDAY SEPTEMBER 16 Chapter 1 Section 1.4 ‐ # 4(a,d), 7(a,c,f), 14; Section 1.5 ‐ # 24, 34, 36; Section 1.6 # 15, 19, 21 1.4 # 4ad Just as in Homework 2 Section 1.3 #12 for something like (a, b+c‐2a, c‐b,c) write it as a(1,‐2,0,0) + b(0,1,‐1,0) + c(0,1,1,1) and recognize the set as all linear combinations of 3 vectors . Remember that ANY span of a collection S of vectors forms a subspace, the span Span(S). Check whether all the vectors are linearly independent, if so it forms a basis, and the number of such vectors is the dimension. The rule for a subspace is that the defining conditions on each entry, such as b + c ‐2a must be a homogeneous linear combination of the variables a,b,c (so no constant term, no product of variables, only constants times variables added up). In these cases you can detect linear independence with the naked eyeball. # 7acf A spanning set for a subspace is a basis iff [if and only if] it is linearly independent. Remember that before U can be a SUBSPACE of a vector space V it must be a SUBSET of V, each element of U must be an element in the set V. # 14 We usually look at an augmented matrix and Gaussian reduction starting with a system of scalar equations AX = B written in rows, forming an augmented matrix and performin...
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