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Consider the vector space V = span ({sin x, cos x}) over R with inner product (, ) : V x V → R defined by (f(x), g(x)) = S f(x)g(x) dx. 1. Find an orthonormal basis B for V with respect to the inner product defined above. 2. Consider the derivative map D: V → V. Determine the adjoint map D* by first deter- mining the matrix [D*]B and then calculating D*(a sin(x) + b cos(x)) where a, bER.

Consider the vector space V = span ({sin x, cos x}) over R with inner product (, ) : V x V → R defined by (f(x), g(x)) = S f(x)g(x) dx. 1. Find an orthonormal basis B for V with respect to the inner product defined above. 2. Consider the derivative map D: V → V. Determine the adjoint map D* by first deter- mining the matrix [D*]B and then calculating D*(a sin(x) + b cos(x)) where a, bER.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 34EQ
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Consider the vector space V = span ({sin x, cos x}) over R with inner product (, ·) : V × V → R defined by (f(x), g(x)) = S™ f(x)g(x) dx. 1. Find an orthonormal basis B for V with respect to the inner product defined above. 2. Consider the derivative map D : V –→ V. Determine the adjoint map D* by first deter- mining the matrix [D*]B_ and then calculating D*(asin(x) + b cos(x)) where a, b e R.
Transcribed Image Text:Consider the vector space V = span ({sin x, cos x}) over R with inner product (, ·) : V × V → R defined by (f(x), g(x)) = S™ f(x)g(x) dx. 1. Find an orthonormal basis B for V with respect to the inner product defined above. 2. Consider the derivative map D : V –→ V. Determine the adjoint map D* by first deter- mining the matrix [D*]B_ and then calculating D*(asin(x) + b cos(x)) where a, b e R.
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