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.Vis not a vector space, and Axioms 7, 8, 9 fail to hold.Vis not a vector space, and Axiom 10 fails to hold.O Vis a vector space.Vis not a vector space, and Axioms 6-10 fail to hold.O Vis not a vector space, and Axiom 6 fails to hold.

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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 operations shown below. x+y=xyAddition cx=xcScalar multiplication If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
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Which vector spaces are isomorphic to R6? a M2,3 b P6 c C[0,6] d M6,1 e P5 f C[3,3] g {(x1,x2,x3,0,x5,x6,x7):xiisarealnumber}
Determine whether the set, together with the standard operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails.M4,6
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Let V be the set of all ordered pairs of real numbers. Check whether V is a vector space over R . If not, state the axiom which fails to hold. (a,b) + (c,d) = (a+c,b+d) and k(a,b) =(ka,0)
Let V denote the set of ordered pairs of real numbers. If (a1, a2) and (b1, b2) are elements of V and c belongs to R, define (a1, a2) + (b1, b2) = (a1 + b1, a2b2) and c(a1, a2) = (ca1, a2). Is V a vector space over R with these operations? Justify your answer.
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.…
Show that if S1 and S2 are arbitrary subsets of a vector space V, then span(S1 ∪ S2) = span(S1)+span(S2).
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Let V be the set of all real numbers, and define vector addition by u ⊕ v = 2(u+v)(for all u,v∈V)and scalar multiplication by c⊙u=cu. Check to see if this combination is a vector space. Check all properties, noting which hold and which do not.
let v be the set of all ordered pairs of real numbers.Check whether V is a vector space over R.If not state the actions which fail to hold (a,b) + (c,d) = (a,b) and k(a,b) =(ka,kb)

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