A (real valued) inner product space is a vector space that has an inner product (x, y) that satisfies the following three axioms: (x, y) = (y,x), (ax, y) = a(x, y), for all real numbers a € R, (x, y) ≥ 0, with equality only if x = 0. Consider the space of bounded functions on the interval [0, 1] with the following inner product (1₁9) = f' 10 f(x)g(x)dx. Bounded just means that the magnitude of both f and g never exceed some fixed large number on the interval [0, 1]. (a) Verify that this space of functions and inner product satisfy the three axioms above. (b) Show that the functions cos(rmx) and cos(anx) for non-negative integers m and n are orthogonal (using the inner product above) for all m n. You may find the following identity useful: cos(a) cos(3) = [cos(a - 3) + cos(a + 3)].
A (real valued) inner product space is a vector space that has an inner product (x, y) that satisfies the following three axioms: (x, y) = (y,x), (ax, y) = a(x, y), for all real numbers a € R, (x, y) ≥ 0, with equality only if x = 0. Consider the space of bounded functions on the interval [0, 1] with the following inner product (1₁9) = f' 10 f(x)g(x)dx. Bounded just means that the magnitude of both f and g never exceed some fixed large number on the interval [0, 1]. (a) Verify that this space of functions and inner product satisfy the three axioms above. (b) Show that the functions cos(rmx) and cos(anx) for non-negative integers m and n are orthogonal (using the inner product above) for all m n. You may find the following identity useful: cos(a) cos(3) = [cos(a - 3) + cos(a + 3)].
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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![A (real valued) inner product space is a vector space that has an inner product \((\mathbf{x}, \mathbf{y})\) that satisfies the following three axioms:
- \(\langle \mathbf{x}, \mathbf{y} \rangle = \langle \mathbf{y}, \mathbf{x} \rangle\),
- \(\langle a\mathbf{x}, \mathbf{y} \rangle = a \langle \mathbf{x}, \mathbf{y} \rangle\), for all real numbers \(a \in \mathbb{R},\)
- \(\langle \mathbf{x}, \mathbf{y} \rangle = 0\), with equality only if \(\mathbf{x} = 0\).
Consider the space of bounded functions on the interval \([0, 1]\) with the following inner product
\[
\langle f, g \rangle = \int_0^1 f(x)g(x) \, dx.
\]
Bounded just means that the magnitude of both \(f\) and \(g\) never exceed some fixed large number on the interval \([0, 1]\).
(a) Verify that this space of functions and inner product satisfy the three axioms above.
(b) Show that the functions \(\cos(\pi mx)\) and \(\cos(\pi nx)\) for non-negative integers \(m\) and \(n\) are orthogonal (using the inner product above) for all \(m \neq n\). You may find the following identity useful: \(\cos(\alpha)\cos(\beta) = \frac{1}{2}[\cos(\alpha - \beta) + \cos(\alpha + \beta)]\).
(c) Plot \(\cos(\pi mx)\) on the interval \([0, 1]\) with \(dx = 0.01\) for \(m = 0, 1, 2, 3, 4,\) and \(5\). Verify numerically using trapezoidal integration (i.e. \texttt{trapz}), that \(\cos(\pi mx)\) and \(\cos(\pi nx)\) are orthogonal for the following \((m, n)\) pairs: (1,4), (2,6), and (3,15).
Note that we have an \text](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6e2f997-9120-4975-9388-a1bc7e4c3a16%2F109d7f1b-8994-41df-84b4-92bd90607bcd%2Fgmobcxn_processed.png&w=3840&q=75)
Transcribed Image Text:A (real valued) inner product space is a vector space that has an inner product \((\mathbf{x}, \mathbf{y})\) that satisfies the following three axioms:
- \(\langle \mathbf{x}, \mathbf{y} \rangle = \langle \mathbf{y}, \mathbf{x} \rangle\),
- \(\langle a\mathbf{x}, \mathbf{y} \rangle = a \langle \mathbf{x}, \mathbf{y} \rangle\), for all real numbers \(a \in \mathbb{R},\)
- \(\langle \mathbf{x}, \mathbf{y} \rangle = 0\), with equality only if \(\mathbf{x} = 0\).
Consider the space of bounded functions on the interval \([0, 1]\) with the following inner product
\[
\langle f, g \rangle = \int_0^1 f(x)g(x) \, dx.
\]
Bounded just means that the magnitude of both \(f\) and \(g\) never exceed some fixed large number on the interval \([0, 1]\).
(a) Verify that this space of functions and inner product satisfy the three axioms above.
(b) Show that the functions \(\cos(\pi mx)\) and \(\cos(\pi nx)\) for non-negative integers \(m\) and \(n\) are orthogonal (using the inner product above) for all \(m \neq n\). You may find the following identity useful: \(\cos(\alpha)\cos(\beta) = \frac{1}{2}[\cos(\alpha - \beta) + \cos(\alpha + \beta)]\).
(c) Plot \(\cos(\pi mx)\) on the interval \([0, 1]\) with \(dx = 0.01\) for \(m = 0, 1, 2, 3, 4,\) and \(5\). Verify numerically using trapezoidal integration (i.e. \texttt{trapz}), that \(\cos(\pi mx)\) and \(\cos(\pi nx)\) are orthogonal for the following \((m, n)\) pairs: (1,4), (2,6), and (3,15).
Note that we have an \text
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