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In Exercises 7–14, either use an appropriate theorem to show that the given set, W, is a
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- 1. Let V be the set of all ordered pairs of real numbers, and consider the following addition and multiplication operations on -() and r-(.2): +Y (M +V+1, +v +1). ki - (ku ku) Is Va vector space? Justify your answerarrow_forward4. Determine whether the given set V is a vector space, justify your answer. (a) V = (x = (1, 22, 23) € R³|*1*2 - 2x3 = 0}. (b) V = (x = (11, 12, 13) € R³|9x1 + 4x2 - 3 = 0}. (c) V = (x = (T1, T2, T3) ER³|3x1-1₂ + 7x3 = 1}. (d) V = (set of all 3 x 3 diagonal matrices}.arrow_forwardQ.1. Let the set H :a² + b² ≥2. Is set H a vector space? Justify your answer. barrow_forward
- 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_forwardSuppose y1 ( x), y2 ( x), y3 ( x) are three different functions of x. The vector space they span could have dimension 1, 2, or 3. Give an example of y1, y2, y3 to show each possibility.arrow_forward41. Vector Spacesarrow_forward
- In 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_forwardVerify that each of the sets in Examples 1– 4 satisfies the axioms fora vector space. Find a basis for each of the vector spaces inExamples 1–4.arrow_forward6. 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_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage