Let T be a linear operator on a real vector space V , and define f : V × V → R by f(x, y) = (x, T y) for all x, y ∈ V . (a) Prove that f is a bilinear form. (b) Prove that f is symmetric if and only if T is self-adjoint. (c) What properties T should have for f to be an inner product on V ? (d) Explain why f may fail to be a bilinear form if V is a complex inner product space.

Let T be a linear operator on a real vector space V , and define f : V × V → R by f(x, y) = (x, T y) for all x, y ∈ V . (a) Prove that f is a bilinear form. (b) Prove that f is symmetric if and only if T is self-adjoint. (c) What properties T should have for f to be an inner product on V ? (d) Explain why f may fail to be a bilinear form if V is a complex inner product space.

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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