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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.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CR: Review Exercises
Problem 54CR: Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector...
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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: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.
(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
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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