7. Let C[-7,7] be the vector space of continuous function over [-n, 1] with an inner product 1 (5,9) = = | f(x)g(x) dæ (a) Show that cos(mx) and sin(nx) are orthogonal for any integers m and n.
7. Let C[-7,7] be the vector space of continuous function over [-n, 1] with an inner product 1 (5,9) = = | f(x)g(x) dæ (a) Show that cos(mx) and sin(nx) are orthogonal for any integers m and n.
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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Question
7
![7. Let C[-7, ] be the vector space of continuous function over [–7, 7] with an inner product
1
(5,9) = - | f(x)g(x) d.x
(a) Show that cos(mx) and sin(nx) are orthogonal for any integers m and n.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb390922a-c60e-4850-aaf9-af48aa8c4d48%2F404cf9fc-a109-458f-b8ac-5d1f8e9c9702%2Fndzfilj_processed.png&w=3840&q=75)
Transcribed Image Text:7. Let C[-7, ] be the vector space of continuous function over [–7, 7] with an inner product
1
(5,9) = - | f(x)g(x) d.x
(a) Show that cos(mx) and sin(nx) are orthogonal for any integers m and n.
Transcribed Image Text:(b) Show that cos(mx) and sin(nx) are unit vectors for any integers m and n.
(c) Compute the vector projection of e" onto cos(mx), where m is an integer.
You may find the following identities helpful:
1
sin(mæ) cos(nx) =;( sin(m – n)a + sin(m + n)x)
1
sin(mx) sin(nx) %3D3( )
cos (m — п)х — cos(m + п)x
COS
2
1
cos(mx) cos(пaх) — %3( cos(m — п)х + сos(m + п)х
e
1+ m² ( cos(mx) +m sin(mæ))
cos(mx) o
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