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product space V to itself to be self-adjoint. (b) If T is self-adjoint, show that if W is any subspace of V which is mapped to itself under T, then there is a complementary subspace W' C V, satisfying V = W OW', which is also preserved under T. (c) Give an example to show that the statement of part (b) ceases to be true in general if the self-adjointness assumption on T is dropped.

product space V to itself to be self-adjoint. (b) If T is self-adjoint, show that if W is any subspace of V which is mapped to itself under T, then there is a complementary subspace W' C V, satisfying V = W OW', which is also preserved under T. (c) Give an example to show that the statement of part (b) ceases to be true in general if the self-adjointness assumption on T is dropped.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. (a) Define what it means for a linear transformation T: V → V from a real inner product space V to itself to be self-adjoint. (b) If T is self-adjoint, show that if W is any subspace of V which is mapped to itself under T, then there is a complementary subspace W' C V, satisfying V = W OW', which is also preserved under T. (c) Give an example to show that the statement of part (b) ceases to be true in general if the self-adjointness assumption on T is dropped.
Transcribed Image Text:4. (a) Define what it means for a linear transformation T: V → V from a real inner product space V to itself to be self-adjoint. (b) If T is self-adjoint, show that if W is any subspace of V which is mapped to itself under T, then there is a complementary subspace W' C V, satisfying V = W OW', which is also preserved under T. (c) Give an example to show that the statement of part (b) ceases to be true in general if the self-adjointness assumption on T is dropped.
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