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- 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.

- 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.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 37E: Let V be the set of all positive real numbers. Determine whether V is a vector space with the...
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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: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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