Let V ={(x, y) x, y = R}. Suppose the vector addition and scalar multiplication in V are definedthe same way as in standard R². Let u = (₁, ₂), v = (v₁, V₂) be vectors in V. We define an innerproduct in V according to the formula(u, v) =1 V2 — U2V₁ + 2U2V2.Show that V equipped with this inner product is a real inner product space.

Answered: Let V = {(x, y) x, y = R}. Suppose the…

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 =
{(x, y) x, y = R}. Suppose the vector addition and scalar multiplication in V are defined
the same way as in standard R². Let u = (₁, ₂), v = (v₁, V₂) be vectors in V. We define an inner
product in V according to the formula
(u, v) =
1 V2 — U2V₁ + 2U2V2.
Show that V equipped with this inner product is a real inner product space.

Expert Solution

Step 1: Given Data

Let $V equals left curly bracket left parenthesis x comma y right parenthesis vertical line space x comma y element of straight real numbers right curly bracket$ .

Let, $u equals left parenthesis u subscript 1 comma u subscript 2 right parenthesis comma space v equals left parenthesis v subscript 1 comma v subscript 2 right parenthesis$ be vectors in V. We define an inner product in V according to the formula:

$less than bold italic u bold comma bold italic v bold greater than equals u subscript 1 v subscript 1 minus u subscript 1 v subscript 2 minus u subscript 2 v subscript 1 plus 2 u subscript 2 v subscript 2$

We have to show that V equipped with this inner product is a real inner product space.