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For Problems 25-31, determine a linearly independent set of vectors that spans the same subspace of V as that spanned by the original set of vectors. 25. V R³, {(1, 2, 3), (−3, 4, 5), (1, -4/3, -5/3)}. 4

For Problems 25-31, determine a linearly independent set of vectors that spans the same subspace of V as that spanned by the original set of vectors. 25. V R³, {(1, 2, 3), (−3, 4, 5), (1, -4/3, -5/3)}. 4

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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For Problems 25-31, determine a linearly independent set of vectors that spans the same subspace of V as that spanned by the original set of vectors. 25. V = 26. V R³, {(3, 1, 5), (0, 0, 0), (1, 2, -1), (-1, 2, 3)}. 27. V = R³, {(1, 1, 1), (1, −1, 1), (1, −3, 1), (3, 1, 2)}. 28. V = R4, {(1, 1, 1, 1), (2, −1, 3, 1), (1, 1, 2, 1), (2, —1, 2, 1)}. = 29. V = R³, {(1, 2, 3), (-3, 4, 5), (1, -4/3, -5/3)}. 30. V = M₂ (R), -1 2 (333) 5 7 12 P₁ (R), {2 — 5x, 3 + 7x, 4 − x}. 31. V = P₂ (R), {2+x², 4 - 2x + 3x², 1+x). For Problems 32-36, use the Wronskian to show that the given functions are linearly independent on the given interval I. 32. f1(x) = 1, ƒ2(x) = x, f3(x) = x², I = (-∞, ∞). 40. Consider the functions f₁(x) = x, f₂(x): ={ = X, -X, (a) Show that f2 is not in C¹ (-∞, ∞). (b) Show that {f1, f2} is linearly dependent on the in- tervals (-∞, 0) and [0, ∞), while it is linearly in- dependent on the interval (-∞, ∞). Justify your results by making a sketch showing both of the functions. if x ≥ 0, if x < 0. X, 41. Determine whether the functions f₁(x) = x, f₂(x) = { if x # 0, if x = 0. = 1, = are linearly dependent or linearly independent on I (-∞, ∞). 42. Show that the functions x-1, if x ≥ 1, f₁(x) = = { 26x² 2(x − 1), if x < 1, f₂(x) = 2x, f3(x) = 3 form a linearly independent set on (-∞, ∞). Determine all intervals on which {f1, f2, f3} is linearly dependent.
Transcribed Image Text:For Problems 25-31, determine a linearly independent set of vectors that spans the same subspace of V as that spanned by the original set of vectors. 25. V = 26. V R³, {(3, 1, 5), (0, 0, 0), (1, 2, -1), (-1, 2, 3)}. 27. V = R³, {(1, 1, 1), (1, −1, 1), (1, −3, 1), (3, 1, 2)}. 28. V = R4, {(1, 1, 1, 1), (2, −1, 3, 1), (1, 1, 2, 1), (2, —1, 2, 1)}. = 29. V = R³, {(1, 2, 3), (-3, 4, 5), (1, -4/3, -5/3)}. 30. V = M₂ (R), -1 2 (333) 5 7 12 P₁ (R), {2 — 5x, 3 + 7x, 4 − x}. 31. V = P₂ (R), {2+x², 4 - 2x + 3x², 1+x). For Problems 32-36, use the Wronskian to show that the given functions are linearly independent on the given interval I. 32. f1(x) = 1, ƒ2(x) = x, f3(x) = x², I = (-∞, ∞). 40. Consider the functions f₁(x) = x, f₂(x): ={ = X, -X, (a) Show that f2 is not in C¹ (-∞, ∞). (b) Show that {f1, f2} is linearly dependent on the in- tervals (-∞, 0) and [0, ∞), while it is linearly in- dependent on the interval (-∞, ∞). Justify your results by making a sketch showing both of the functions. if x ≥ 0, if x < 0. X, 41. Determine whether the functions f₁(x) = x, f₂(x) = { if x # 0, if x = 0. = 1, = are linearly dependent or linearly independent on I (-∞, ∞). 42. Show that the functions x-1, if x ≥ 1, f₁(x) = = { 26x² 2(x − 1), if x < 1, f₂(x) = 2x, f3(x) = 3 form a linearly independent set on (-∞, ∞). Determine all intervals on which {f1, f2, f3} is linearly dependent.
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