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).

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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).
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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