(3) Let V be R2, the set of all ordered pairs (x, y) of real numbers. Define an operation of"addition" by(u, v)(x, y) = (u+x+1,v+y+1)for all (u, v) and (x, y) in V. Define an operation of "scalar multiplication" bya(r, y) = (ar, ay)for all a ER and (x, y) = V.Under the operationsand the set V is not a vector space. The vector spaceaxioms (see 5.1.1 (1)-(10)) which fail to hold are and

Given, is the set of all ordered pairs of real numbers.The operation of addition is defined by,…

Answered: (3) Let V be R², the set of all ordered…

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

(3) Let V be R2, the set of all ordered pairs (x, y) of real numbers. Define an operation of
"addition" by
(u, v)
(x, y) = (u+x+1,v+y+1)
for all (u, v) and (x, y) in V. Define an operation of "scalar multiplication" by
a(r, y) = (ar, ay)
for all a ER and (x, y) = V.
Under the operations
and the set V is not a vector space. The vector space
axioms (see 5.1.1 (1)-(10)) which fail to hold are and

Expert Solution

Step 1: Introduction

Given,

$V$ is the set of all ordered pairs $left parenthesis x comma space y right parenthesis$ of real numbers.

The operation of addition is defined by,

$left parenthesis u comma space v right parenthesis ⊞ space left parenthesis x comma space y right parenthesis equals left parenthesis u plus x plus 1 comma space v plus y plus 1 right parenthesis$ for all $left parenthesis u comma space v right parenthesis$ and $left parenthesis x comma space y right parenthesis$ in $V$ .

The operation of scalar multiplication is defined by,

$alpha space ⊡ left parenthesis x comma space y right parenthesis equals left parenthesis alpha x comma space alpha y right parenthesis$ for all $alpha element of straight real numbers$ and $left parenthesis x comma space y right parenthesis element of V$ .

Aim: To find which axioms of the vector space fail to hold.

A set $V$ with two vector operations $plus comma space.$ is said to be a vector space over the field $F$ if the following conditions hold:

$left parenthesis i right parenthesis space alpha element of V comma space beta element of V rightwards double arrow alpha plus space beta element of V. left parenthesis i i right parenthesis space left parenthesis alpha plus space beta right parenthesis plus gamma equals alpha plus space left parenthesis beta plus gamma right parenthesis comma space for all space alpha comma space beta comma space gamma element of V.$

$left parenthesis i i i right parenthesis space there exists space e element of V space$ such that $space alpha plus space e equals alpha.$

(iv) For every $alpha element of V comma space there exists space beta element of V space$ such that $alpha plus beta equals e$ .

(v) $alpha plus beta equals beta plus alpha space comma space for all alpha comma beta element of V$

$left parenthesis v i right parenthesis space c element of F comma space beta element of V rightwards double arrow c beta element of V. left parenthesis v i i right parenthesis space c left parenthesis alpha plus space beta right parenthesis equals c alpha plus space c beta plus gamma right parenthesis comma space for all space alpha comma space beta element of V comma space c element of F. left parenthesis v i i i right parenthesis space c left parenthesis space d alpha right parenthesis equals left parenthesis c d right parenthesis alpha comma space for all space alpha element of V comma space c comma space d element of F left parenthesis i x right parenthesis space left parenthesis c plus d right parenthesis alpha equals c alpha plus d alpha comma space for all space alpha element of V comma space c comma space d element of F left parenthesis x right parenthesis space 1 space alpha equals space alpha$