Recall the inner product (ƒ, g) = [”, f(z)g(z) dz defined on the vector space C([-]) of continuous (real-valued) functions ƒ : [-, π] → R. Observe the following integral formulas, which are valid for natural numbers m, n ≥ 1: • sin(mz) cos(nx) dx = 0 (0 cos(mz) cos(nx) dx = [”, sin(m²) sin(nx) dx = ‹ ifm #n la ifm=n 1. What do these formulas say about the angles between the vectors sin(n.), sin(m.), cos(n.) and cos(m.) in the space C([-])? What do say you about the lengths of these vectors? 2. Do you think these vectors form a basis for the space C([-])? Why or why not?
Recall the inner product (ƒ, g) = [”, f(z)g(z) dz defined on the vector space C([-]) of continuous (real-valued) functions ƒ : [-, π] → R. Observe the following integral formulas, which are valid for natural numbers m, n ≥ 1: • sin(mz) cos(nx) dx = 0 (0 cos(mz) cos(nx) dx = [”, sin(m²) sin(nx) dx = ‹ ifm #n la ifm=n 1. What do these formulas say about the angles between the vectors sin(n.), sin(m.), cos(n.) and cos(m.) in the space C([-])? What do say you about the lengths of these vectors? 2. Do you think these vectors form a basis for the space C([-])? Why or why not?
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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![Recall the inner product (f,g) = S, f(x)g(x) dx defined on the vector space C([-]) of continuous (real-valued)
functions f : [-, π] → R. Observe the following integral formulas, which are valid for natural numbers m, n > 1:
• sin(ma) cos(nx) dx = 0
(0
cos(mx) cos(nx) dx = sin(mx) sin(nx) dx =
ifm #n
| if m=n
1. What do these formulas say about the angles between the vectors sin(n.), sin(m.), cos(n.) and cos(m.) in the space
C([-])? What do say you about the lengths of these vectors?
2. Do you think these vectors form a basis for the space C([-,])? Why or why not?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e8ed93-7bef-4409-89ed-52264f64a27e%2Fca1ef2c2-e35f-4396-b713-0551d89487bd%2F24lowmh_processed.png&w=3840&q=75)
Transcribed Image Text:Recall the inner product (f,g) = S, f(x)g(x) dx defined on the vector space C([-]) of continuous (real-valued)
functions f : [-, π] → R. Observe the following integral formulas, which are valid for natural numbers m, n > 1:
• sin(ma) cos(nx) dx = 0
(0
cos(mx) cos(nx) dx = sin(mx) sin(nx) dx =
ifm #n
| if m=n
1. What do these formulas say about the angles between the vectors sin(n.), sin(m.), cos(n.) and cos(m.) in the space
C([-])? What do say you about the lengths of these vectors?
2. Do you think these vectors form a basis for the space C([-,])? Why or why not?
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