problemset2 - MAS 213 Linear Algebra II Problem list for...

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MAS 213: Linear Algebra II. Problem list for Week #2. Tutorial on 14th September. This week’s topics: The span of a set. Linear dependence and independence. Tutorial problems: Problem 1: (Problems 1.4.3(e), 1.4.4(c) in [FIS].) In the following parts you are given a vector space V , a vector v in it, and a subset S of the vector space. Determine if the given vector v lies in the span of the given subset S . If it does, find an explicit representation of v as a linear combination of the vectors in S . 1. V = R 3 , v = (5 , 1 , - 5), and S = { (1 , - 2 , - 3) , ( - 2 , 3 , - 4) } . 2. V = P 3 ( R ), v = - 2 x 3 - 11 x 2 + 3 x + 2, and S = { x 3 - 2 x 2 + 3 x - 1 , 2 x 3 + x 2 + 3 x - 2 } . 3. V = M 2 × 2 ( R ), v = [ 2 - 1 1 3 ] , and S = {[ 1 1 1 1 ] , [ 0 1 1 1 ] , [ 0 0 1 1 ]} . Problem 2: (Problem 1.4.6 in [FIS].) Consider the following subset of R 3 : S = { (1 , 1 , 1) , (1 , 1 , 0) , (1 , 0 , 0) } . Prove that span( S ) = R 3 . 1
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Problem 3: (Problem 1.5.2 parts (c), (d), (e) and (f) in [FIS].) In the following parts you are given a vector space V and a subset S of it. Determine if the given subset is linearly dependent or independent.
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