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Let V be the set of all ordered pairs of real numbers under the following operations of addition 1. and scalar multiplication. Define addition and scalar multiplication on V as follows: (a, b) + (c,d) = (a +c+2, b+d-5) and k(a, b) = (ka, kb) a. There exists a zero vector 0 in V such that for any v in V, 0 + v = v + 0 = v. Which ordered pair in V is the zero vector? b. V is not a vector space. Determine one of the vector space axioms that is not satisfied by V, and demonstrate with a counterexample.

Let V be the set of all ordered pairs of real numbers under the following operations of addition 1. and scalar multiplication. Define addition and scalar multiplication on V as follows: (a, b) + (c,d) = (a +c+2, b+d-5) and k(a, b) = (ka, kb) a. There exists a zero vector 0 in V such that for any v in V, 0 + v = v + 0 = v. Which ordered pair in V is the zero vector? b. V is not a vector space. Determine one of the vector space axioms that is not satisfied by V, and demonstrate with a counterexample.

Advanced Engineering Mathematics
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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1. Let V be the set of all ordered pairs of real numbers under the following operations of addition and scalar multiplication. Define addition and scalar multiplication on V as follows: (a, b) + (c,d) = (a +c+2, b + d-5) and k(a, b) = (ka, kb) a. There exists a zero vector 0 in V such that for any v in V, 0 + v = v + 0 = v. Which ordered pair in V is the zero vector? b. V is not a vector space. Determine one of the vector space axioms that is not satisfied by V, and demonstrate with a counterexample.
Transcribed Image Text:1. Let V be the set of all ordered pairs of real numbers under the following operations of addition and scalar multiplication. Define addition and scalar multiplication on V as follows: (a, b) + (c,d) = (a +c+2, b + d-5) and k(a, b) = (ka, kb) a. There exists a zero vector 0 in V such that for any v in V, 0 + v = v + 0 = v. Which ordered pair in V is the zero vector? b. V is not a vector space. Determine one of the vector space axioms that is not satisfied by V, and demonstrate with a counterexample.
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