6. (a) Let T: L²[0, 2π] → L²[0, 2π] be given by 2п Tf(x) = S cos(xt) f(t) dt. Show that (i) T is self-adjoint. (ii) cos x and sin x are eigenvectors of T.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...
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6.
(a)
Let T : L²[0, 2π] → L²[0, 2ñ] be given by
2π
Tf(x) =
s
cos(xt) f(t) dt.
Show that
(i) T is self-adjoint.
(ii) cos x and sin x are eigenvectors of T.
Transcribed Image Text:6. (a) Let T : L²[0, 2π] → L²[0, 2ñ] be given by 2π Tf(x) = s cos(xt) f(t) dt. Show that (i) T is self-adjoint. (ii) cos x and sin x are eigenvectors of T.
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