1. Suppose V is the vector space of continuous functions f : [-1, 1] → R. (a) Show that (,): V × V → R given by (f,g) = [' , ƒ(x)g(x)(1 — x²) dæ defines an inner product. (b) Let p(x) = 1 and q(x) = x be elements of V. (i) Calculate the value of ||p|| (with respect to the given inner product). (ii) Calculate the value of ||q|| (with respect to the given inner product). (iii) Calculate the angle between p and q (with respect to the given inner product).
1. Suppose V is the vector space of continuous functions f : [-1, 1] → R. (a) Show that (,): V × V → R given by (f,g) = [' , ƒ(x)g(x)(1 — x²) dæ defines an inner product. (b) Let p(x) = 1 and q(x) = x be elements of V. (i) Calculate the value of ||p|| (with respect to the given inner product). (ii) Calculate the value of ||q|| (with respect to the given inner product). (iii) Calculate the angle between p and q (with respect to the given inner product).
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.4: Spanning Sets And Linear Independence
Problem 76E: Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are...
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![1. Suppose V is the vector space of continuous functions ƒ : [-1, 1] → R.
(a) Show that (,): V x V → R given by
defines an inner product.
(f,g) = ['ª¸ ƒ(x)g(x)(1 – xa²) dx
1
(b) Let p(x) = 1 and q(x) = x be elements of V.
(i) Calculate the value of ||p|| (with respect to the given inner product).
(ii) Calculate the value of ||q|| (with respect to the given inner product).
(iii) Calculate the angle between p and q (with respect to the given inner product).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a97bb2c-6d83-4604-9a70-f559a7370826%2F92256f5a-b764-4e43-a41d-d12d49b4f987%2Fyw3yvze_processed.png&w=3840&q=75)
Transcribed Image Text:1. Suppose V is the vector space of continuous functions ƒ : [-1, 1] → R.
(a) Show that (,): V x V → R given by
defines an inner product.
(f,g) = ['ª¸ ƒ(x)g(x)(1 – xa²) dx
1
(b) Let p(x) = 1 and q(x) = x be elements of V.
(i) Calculate the value of ||p|| (with respect to the given inner product).
(ii) Calculate the value of ||q|| (with respect to the given inner product).
(iii) Calculate the angle between p and q (with respect to the given inner product).
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