Let B= {1, cost, cos?t, .. cos t} and C={1, cost, cos 2t, ... , cos 61). Assume the following trigonometric identities. cos 2t = - 1+2 cos²t cos 3t = - 3 cost+ 4 cost cos 4t = 1-8 cos t+8 cos t cos 5t = 5 cost- 20 cost+ 16 cos St cos 6t = - 1+ 18 cos²t- 48 cos t+ 32 cos ºt Let H be the subspace of functions spanned by the functions in B. Then Bis a basis for H. Complete parts (a) and (b). a. Write the B-coordinate vectors of the vectors in C as the columns of a matrix P, and use them to show that C is a linearly independent set in H. P=O
Let B= {1, cost, cos?t, .. cos t} and C={1, cost, cos 2t, ... , cos 61). Assume the following trigonometric identities. cos 2t = - 1+2 cos²t cos 3t = - 3 cost+ 4 cost cos 4t = 1-8 cos t+8 cos t cos 5t = 5 cost- 20 cost+ 16 cos St cos 6t = - 1+ 18 cos²t- 48 cos t+ 32 cos ºt Let H be the subspace of functions spanned by the functions in B. Then Bis a basis for H. Complete parts (a) and (b). a. Write the B-coordinate vectors of the vectors in C as the columns of a matrix P, and use them to show that C is a linearly independent set in H. P=O
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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Transcribed Image Text:Let B= {1, cost, cos?t, ., cos °t} and C={1, cost, cos 2t, ... , cos 6t). Assume the following trigonometric identities.
cos 2t = - 1+2 cos ?t
cos 3t = - 3 cos t+ 4 cos t
cos 41 = 1-8 cos?t+8 cos t
cos 5t = 5 cost- 20 cos t+ 16 cos St
cos 61 = - 1+ 18 cos?t- 48 cos "t+ 32 cos °t
Let H be the subspace of functions spanned by the functions in B. Then B is a basis for H. Complete parts (a) and (b).
a. Write the B-coordinate vectors of the vectors in C as the columns of a matrix P, and use them to show that C is a linearly independent set in H.
P =
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