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9. Consider the vector space V of all C² functions u : [0, L] → R satisfying the boundary conditions u'(0) = 0 and u(L) = 0, and the linear transformation Mu= u"(x). (a) Show that for any u, v € V, (Mu, v) = (u, Mv). [Hint: integrate by parts twice, and use the boundary conditions.] (b) Suppose A, μ are eigenvalues of M, so there exist nontrivial functions u, v € V, with Mu=u"(x) = \u(x) and Mv = v″(x) = µv. Show that if \‡ µ, then (u, v) = 0. (c) Show that {cos ( OS (2k-1)лx 2L (2k-1) − ((²²−1)ª)², and conclude that they form an orthogonal family in --} distinct eigenvalues λ = −( V. : k = 1, 2, 3, .. are eigenfunctions of M with

9. Consider the vector space V of all C² functions u : [0, L] → R satisfying the boundary conditions u'(0) = 0 and u(L) = 0, and the linear transformation Mu= u"(x). (a) Show that for any u, v € V, (Mu, v) = (u, Mv). [Hint: integrate by parts twice, and use the boundary conditions.] (b) Suppose A, μ are eigenvalues of M, so there exist nontrivial functions u, v € V, with Mu=u"(x) = \u(x) and Mv = v″(x) = µv. Show that if \‡ µ, then (u, v) = 0. (c) Show that {cos ( OS (2k-1)лx 2L (2k-1) − ((²²−1)ª)², and conclude that they form an orthogonal family in --} distinct eigenvalues λ = −( V. : k = 1, 2, 3, .. are eigenfunctions of M with

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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9. Consider the vector space V of all C² functions u: [0, L] → R satisfying the boundary conditions u'(0) = 0 and u(L) = 0, and the linear transformation Mu = u"(x). (a) Show that for any u, v E V, (Mu, v) = (u, Mv). [Hint: integrate by parts twice, and use the boundary conditions.] (b) Suppose X, μ are eigenvalues of M, so there exist nontrivial functions u, v € V, with Muu"(x) = Xu(x) and Mv = v"(x) = µv. Show that if X μ, then (u, v) = 0. (c) Show that {cos (12% -1) rr): : k = 1,2,3,.. }} 2L are eigenfunctions of M with -((2k-1))², and conclude that they form an orthogonal family in 2L distinct eigenvalues X V.
Transcribed Image Text:9. Consider the vector space V of all C² functions u: [0, L] → R satisfying the boundary conditions u'(0) = 0 and u(L) = 0, and the linear transformation Mu = u"(x). (a) Show that for any u, v E V, (Mu, v) = (u, Mv). [Hint: integrate by parts twice, and use the boundary conditions.] (b) Suppose X, μ are eigenvalues of M, so there exist nontrivial functions u, v € V, with Muu"(x) = Xu(x) and Mv = v"(x) = µv. Show that if X μ, then (u, v) = 0. (c) Show that {cos (12% -1) rr): : k = 1,2,3,.. }} 2L are eigenfunctions of M with -((2k-1))², and conclude that they form an orthogonal family in 2L distinct eigenvalues X V.
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