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In Exercises 7–14, either use an appropriate theorem to show that the given set, W, is a
14.
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- 1.W.4 We'll work inside the vector space of polynomials in degree < 2, which is denoted P<2. Let P1 = 1, p2 = x + 2, and p3 = (x + 2)². Leť's think about Span{p1, P2, P3}. a) What polynomial is 3p1 + 2p2 – P3? b) I claim that a = a¡P1 + a2P2 + azp3 for some coefficients a1, a2, az in R. Find a1, a2, az. Hint: az = 0. c) I claim that a? = bịPi + b2p2 + b3p3 for some coefficients b1, b2, bz in R. Find b1, b2, bz. Hint: This problem isn't quite so cut and dry. Try to find three equations, one for each coefficient in the polynomial, and solve them for b1, b2, b3.arrow_forwardThe given set together with the given operations is not a vector space. List the properties that fail to hold.arrow_forwardConsider the empty set, i.e. X={ }. Which property of vector space is not satisfied and how?arrow_forward
- Suppose that {x1, x2, x3} is a linearly independent set of vectors in a vector space V .Must {x1, x1 + 3x2, x1 + 3x2 + 5x3} be a linearly independent set of vectors? Explain.arrow_forwardWhat is the solution to the linear algebra problem that is in the attached image? Does it involve the span(F)?arrow_forwardIn addition: indicate which vector space axioms V satisifies and which, if any, it does not. Show a step by step proof of each axiom. Picture of questions is attached.arrow_forward
- 6. Can any of the given sets of 3-vectors below span the 3-space? Why or why not? (a) [1 2 1] [231] [3 4 2] (b) [8 1 3] [128] [-7 1 5]arrow_forwardLet B={(1,-2,1), (4,-7,5), (5,-8,8)), and x=(-6,10,-7) Find [x]B Give your answer in the form (a,b,c) with no spacesarrow_forward1) Suppose R is the set of real numbers, and we define addition (using the symbol ) as: x y = max(x,y) (the maximum of x and y) with scalar multiplication having the normal definition. Is R vector space with these operations? Show whether or not all 8 axioms hold for these definitions.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage