Determine whether the set equipped with the given operations is a vector space.For those that are not vector spaces identify the vector space axioms that fail.The set of all real numbers x with the standard operations of addition and multiplication.O Visa vector space.V is not a vector space, and Axiom 10 fails to hold.O Vis not a vector space, and Axioms 6 - 10 fail to hold.O Vis not a vector space, and Axioms 7, 8, 9 fail to hold.O Vis not a vector space, and Axiom 6 fails to hold.

Consider the set of real numbers ℝℝ with the standard operations of addition and multiplication.…

Answered: Determine whether the set equipped with…

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 positive real numbers. Determine whether V is a vector space with the following operations. x + y = xy Addition cx = xc Scalar multiplication If it is, then verify each vector space axiom; if it is not, then state all vector space axioms that fail. STEP 1: Check each of the 10 axioms.(1) u + v is in V. This axiom holds.This axiom fails. (2) u + v = v + u This axiom holds.This axiom fails. (3) u + (v + w) = (u + v) + w This axiom holds.This axiom fails. (4) V has a zero vector 0 such that for every u in V, u + 0 = u. This axiom holds.This axiom fails. (5) For every u in V, there is a vector in V denoted by −u such that u + (−u) = 0. This axiom holds.This axiom fails. (6) cu is in V. This axiom holds.This axiom fails. (7) c(u + v) = cu + cv This axiom holds.This axiom fails. (8) (c + d)u = cu + du This axiom holds.This axiom fails. (9) c(du) = (cd)u This axiom holds.This axiom fails.…

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Determine whether the set equipped with the given operations is a vector space. For those that are not vector spaces identify the vector space axioms that fail. The set of all real numbers x with the standard operations of addition and multiplication.