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601-hw-i - MATH 601 AUTUMN 2008 Additional homework I...

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MATH 601, AUTUMN 2008 Additional homework I, October 2 PROBLEMS 1. Determine if the set X consisting of the following three functions t - 1 , t 2 + 3 , t - t 2 of the real variable t , spans the whole real vector space V of real-valued polynomials in t of degree not exceeding 2. 2. Let W be the subset of the complex vector space C 4 formed by all quadruples ( x, y, z, u ) with x - y + 3 u = 2 z + 7 u = 0. Is W a subspace? If the answer is ‘yes’, find a finite set Y with W = Span ( Y ). 3. Let V denote the complex vector space of polynomials in the complex variable z whose degree does not exceed 4. Is the two-element system Y = { z 2 - z + 1 , z 4 + z } linearly independent? If the answer is ‘yes’, find a basis X of V containing Y . 4. Use successive elementary modifications to replace the system Y = { (2 , 1 , 5) , (1 , 1 , 2) , (3 , 4 , 0) } in R 3 with an equivalent system Y containing the system X = { (1 , 1 , 1) , (2 , 3 , 4) } . 5. Find the dimension of the real space V of all C functions f : R R with f (5) = 0 identically and f (1) = f ( - 1) = 0. (Why is V a vector space?)
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MATH 601, AUTUMN 2008 SOLUTIONS,
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