Let V be a finite-dimensional inner product space over C with inner product (, ), and let a, b e V \0. Define T : V +V by T(v) = (v, a) b for all v e V. (i) Show that T is a linear map. (ii) For v E V, find T*(v) in terms of v, a, b. (iii) Prove that if T = T*, then b = la for some A E R.

Let V be a finite-dimensional inner product space over C with inner product (, ), and let a, b e V \0. Define T : V +V by T(v) = (v, a) b for all v e V. (i) Show that T is a linear map. (ii) For v E V, find T(v) in terms of v, a, b. (iii) Prove that if T = T, then b = la for some A E R.

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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Question

Let V be a finite-dimensional inner product space over C with inner product (, ), and let
a, b e V \0. Define T: V + V by
T(v) = (v, a) b
for all v E V.
(i) Show that T is a linear map.
(ii) For v E V, find T*(v) in terms of v, a, b.
(iii) Prove that if T = T*, then b = Xa for some A E R.

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