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p₁(x) = 3x +5x², P₂(x)=1+x+x², P3(x)=2x, P4(x) = 1 + 2x².

p₁(x) = 3x +5x², P₂(x)=1+x+x², P3(x)=2x, P4(x) = 1 + 2x².

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve #22 and show step by step and explain each step, POST PICTURES OF YOUR WORK.

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For Problems 19-22, determine whether the given set of vec- tors is linearly independent in P₂(R). 19. p₁(x) = 1-x, p2(x) = 1+x. 20. p1(x) = 2+3x, p2(x) = 4 + 6x. 21. p₁(x) = 13x², p₂(x) = 2x + x², P3(x) = 5. 22. p₁(x) = 3x +5x², p₂(x)=1+x+x²₂ P3(x) = 2x, p4(x) = 1+ 2x². 23. Show that the vectors P₁(x) = a + bx and p₂(x) = c+dx are linearly independent in P₁ (R) if and only if the constants a, b, c, d satisfy ad — bc ‡ 0. - 24. If f1(x) = cos 2x, f2(x) sin² x, f(x) = cos²x, determine whether {f1, f2, f3} is linearly dependent or linearly independent in C(-∞, ∞). For Problems 25-31, determine a linearly independent set of = 4.5 Linear Dependence and Linear Independence 297 35. f₁(x) = e²x, ƒ2(x) = e³x, ƒ3(x) = e¯x, I = (-∞, ∞). 36. fi (x) = x², if x ≥ 0, = f2(x) = 7x²,1 = (-∞, ∞0). For Problems 37-39, show that the Wronskian of the given functions is identically zero on (-∞, ∞). Determine whether the functions are linearly independent or linearly dependent on that interval. 37. f₁(x) = 1, ƒ2(x) = x, ƒ3(x) = 2x – 1. 38. f1(x)=e*, f2(x) = e¯*, f3(x) = coshx. 39. f1(x) = 2x³, if x ≥ 0, f2(x) = { _3x²³, if x < 0, 40. Consider the functions f₁(x) = x,
Transcribed Image Text:For Problems 19-22, determine whether the given set of vec- tors is linearly independent in P₂(R). 19. p₁(x) = 1-x, p2(x) = 1+x. 20. p1(x) = 2+3x, p2(x) = 4 + 6x. 21. p₁(x) = 13x², p₂(x) = 2x + x², P3(x) = 5. 22. p₁(x) = 3x +5x², p₂(x)=1+x+x²₂ P3(x) = 2x, p4(x) = 1+ 2x². 23. Show that the vectors P₁(x) = a + bx and p₂(x) = c+dx are linearly independent in P₁ (R) if and only if the constants a, b, c, d satisfy ad — bc ‡ 0. - 24. If f1(x) = cos 2x, f2(x) sin² x, f(x) = cos²x, determine whether {f1, f2, f3} is linearly dependent or linearly independent in C(-∞, ∞). For Problems 25-31, determine a linearly independent set of = 4.5 Linear Dependence and Linear Independence 297 35. f₁(x) = e²x, ƒ2(x) = e³x, ƒ3(x) = e¯x, I = (-∞, ∞). 36. fi (x) = x², if x ≥ 0, = f2(x) = 7x²,1 = (-∞, ∞0). For Problems 37-39, show that the Wronskian of the given functions is identically zero on (-∞, ∞). Determine whether the functions are linearly independent or linearly dependent on that interval. 37. f₁(x) = 1, ƒ2(x) = x, ƒ3(x) = 2x – 1. 38. f1(x)=e*, f2(x) = e¯*, f3(x) = coshx. 39. f1(x) = 2x³, if x ≥ 0, f2(x) = { _3x²³, if x < 0, 40. Consider the functions f₁(x) = x,
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