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Exercise 2.8 Are the following sets of functions linearly independent or dependent on the positive x-axis? (a) x, x+2, x +5 (b) cos x, cos 2x, cos 3x (c) sin x, cos x, sin 2x

Exercise 2.8 Are the following sets of functions linearly independent or dependent on the positive x-axis? (a) x, x+2, x +5 (b) cos x, cos 2x, cos 3x (c) sin x, cos x, sin 2x

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 excercise 2.8
Are the following sets of functions linearly independent or dependent? (a) 4e, e, Se (b) cos x, e, 3 sin x (e) 7, x2, 9x, e- Exercise 2.8 Are the following sets of functions linearly independent or dependent on the positive -axis? (a) x, x+2, x +5 (b) cos x, cos 2x, cos 3x (c) sin x, cos-x, sin 2x (d) x, (x-1)2, (x + 1) (e) sinh x, cosh² x, 1 Exercise 2.9 Are the following sets of vectors linearly independent or dependent over the complex field? (a) (2,-3, 0), (0, 0, 1), (2i, i, -1) (b) (0, 4, 0), (i, -3i, i), (2, 0, 1) (c) (i, 1, 2), (3, i,-1), (-i,3i, 5i) Exercise 2.10 Are the following sets of vectors (in the three-dimensional Euclidean space) linearly indepe dent or dependent? (a) (4, 5, 6), (1, 2, 3), (7, 8, 9) (b) (1, 0, 0), (Q, –5, 0), (0,0, /7) (c) (5, 4, 1),(2, 0, -2), (0, 6, –1) 2 is also a pre Exercise 2.11 Show that if A is a projection operator, the operator ) ator. dess of Exercise 2.12 Show that y)(pw1/lw)is a projection ope 15 nomal
Transcribed Image Text:Are the following sets of functions linearly independent or dependent? (a) 4e, e, Se (b) cos x, e, 3 sin x (e) 7, x2, 9x, e- Exercise 2.8 Are the following sets of functions linearly independent or dependent on the positive -axis? (a) x, x+2, x +5 (b) cos x, cos 2x, cos 3x (c) sin x, cos-x, sin 2x (d) x, (x-1)2, (x + 1) (e) sinh x, cosh² x, 1 Exercise 2.9 Are the following sets of vectors linearly independent or dependent over the complex field? (a) (2,-3, 0), (0, 0, 1), (2i, i, -1) (b) (0, 4, 0), (i, -3i, i), (2, 0, 1) (c) (i, 1, 2), (3, i,-1), (-i,3i, 5i) Exercise 2.10 Are the following sets of vectors (in the three-dimensional Euclidean space) linearly indepe dent or dependent? (a) (4, 5, 6), (1, 2, 3), (7, 8, 9) (b) (1, 0, 0), (Q, –5, 0), (0,0, /7) (c) (5, 4, 1),(2, 0, -2), (0, 6, –1) 2 is also a pre Exercise 2.11 Show that if A is a projection operator, the operator ) ator. dess of Exercise 2.12 Show that y)(pw1/lw)is a projection ope 15 nomal
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