a) If T #0 is a nilpotent operator on V, show that there existsve V such that Tv # 0 but T²u = 0.b) Suppose S+ I is an isometry on the inner product space V. If S is also self-adjointshow that -1 is an eigenvalue of S.

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a) If T # 0 is a nilpotent operator on V, show that there exists veV such that Tv +0 but Tv = 0. b) Suppose S I is an isometry on the inner product space V. If S is also self-adjoint show that -1 is an eigenvalue of S.

a) If T # 0 is a nilpotent operator on V, show that there exists veV such that Tv +0 but Tv = 0. b) Suppose S I is an isometry on the inner product space V. If S is also self-adjoint show that -1 is an eigenvalue of S.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Question

a) If T #0 is a nilpotent operator on V, show that there exists
ve V such that Tv # 0 but T²u = 0.
b) Suppose S+ I is an isometry on the inner product space V. If S is also self-adjoint
show that -1 is an eigenvalue of S.

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