()--)U1V1Let V = C², and for u =E V,defineV =U2V2(u, v) = u101 + iuj02 – iuzī1 + 2u202.(i) Show that this is an inner product on V.(ii) Find an orthonormal basis of V with respect to this inner product.(iii) Define a linear map T: V → V by2 0T(v)(v E V).0 1= Av for all V E V, where T* is theFind a complex 2 × 2 matrix A such that T*(v)adjoint of T with respect to the above inner product.

Step:-1 Note:- Let V be a vector space and a, b, c∈V then an inner product , on V is a operation…

()--) U1 Let V = C2, and for u = V1 V = E V, define %3| U2 V2 (u, v) = u101 + iujū2 – iuzī1 + 2uzī2. i) Show that this is an inner product on V. ii) Find an orthonormal basis of V with respect to this inner product.

()--) U1 Let V = C2, and for u = V1 V = E V, define %3| U2 V2 (u, v) = u101 + iujū2 – iuzī1 + 2uzī2. i) Show that this is an inner product on V. ii) Find an orthonormal basis of V with respect to this inner product.

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

()--)
U1
V1
Let V = C², and for u =
E V,
define
V =
U2
V2
(u, v) = u101 + iuj02 – iuzī1 + 2u202.
(i) Show that this is an inner product on V.
(ii) Find an orthonormal basis of V with respect to this inner product.
(iii) Define a linear map T: V → V by
2 0
T(v)
(v E V).
0 1
= Av for all V E V, where T* is the
Find a complex 2 × 2 matrix A such that T*(v)
adjoint of T with respect to the above inner product.

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