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= 5.8. Let f₁(x) = x³ x² +1, f₂(x) = x³ + x² + x and f3 (x) = x³ + 2x²+3x+1 be three polynomials in the vector space P3 and let W Span{f(x), f₂(x), ƒ3 (x)}. Does the polynomial g(x) = 4x³ + 11x² + 11x belong to the subspace W? If yes, write g(x) as a linear combination of the vectors f₁(x), f₂ (x), f(x).

= 5.8. Let f₁(x) = x³ x² +1, f₂(x) = x³ + x² + x and f3 (x) = x³ + 2x²+3x+1 be three polynomials in the vector space P3 and let W Span{f(x), f₂(x), ƒ3 (x)}. Does the polynomial g(x) = 4x³ + 11x² + 11x belong to the subspace W? If yes, write g(x) as a linear combination of the vectors f₁(x), f₂ (x), f(x).

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.CM: Cumulative Review
Problem 16CM
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.3 5.8. Let f₁(x) = x³ -x² +1, f₂(x) = x³ + x²+x and f3 (x) = x³ +2x²+3x+1 be three polynomials in the vector space P3 and let W = Span{f₁(x), f₂(x), f(x)}. Does the polynomial g(x) = 4x³ + 11x² + 11x belong to the subspace W? If yes, write g(x) as a linear combination of the vectors f₁(x), f₂ (x), ƒ3 (x).
Transcribed Image Text:.3 5.8. Let f₁(x) = x³ -x² +1, f₂(x) = x³ + x²+x and f3 (x) = x³ +2x²+3x+1 be three polynomials in the vector space P3 and let W = Span{f₁(x), f₂(x), f(x)}. Does the polynomial g(x) = 4x³ + 11x² + 11x belong to the subspace W? If yes, write g(x) as a linear combination of the vectors f₁(x), f₂ (x), ƒ3 (x).
5.8 Let g(x) = C₁f₁(x) + C₂f₂(x) + C3f3(x), for all x. Simplify this to get (C₁+C₂ + C3-4)x³ + (-C₁ +6₂ + 2C3-11)x² + (C₂+3c3 - 11)x+ C₁ + C3 = 0, for all x. Since 1, x, x², x³ are linearly independent we must have C₁ + C₂ + C3 - 4 = 0, −C₁ + C₂ + 2C3 - 11 = 0, C₂ + 3C3 - 11, C₁ + C3 = 0. 7 Solve to get c₁ = ₁, C₂ = 4, C3 = C2 3 Hence, g(x) is in W and g(x) = -f(x) + 4ƒ₂ (x) + f(x). ——
Transcribed Image Text:5.8 Let g(x) = C₁f₁(x) + C₂f₂(x) + C3f3(x), for all x. Simplify this to get (C₁+C₂ + C3-4)x³ + (-C₁ +6₂ + 2C3-11)x² + (C₂+3c3 - 11)x+ C₁ + C3 = 0, for all x. Since 1, x, x², x³ are linearly independent we must have C₁ + C₂ + C3 - 4 = 0, −C₁ + C₂ + 2C3 - 11 = 0, C₂ + 3C3 - 11, C₁ + C3 = 0. 7 Solve to get c₁ = ₁, C₂ = 4, C3 = C2 3 Hence, g(x) is in W and g(x) = -f(x) + 4ƒ₂ (x) + f(x). ——
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