(6)cu is in V.This axiom holds.O This axiom fails.(7)c(u + v) = cu + cvThis axiom holds.O This axiom fails.(c + d)u = cu + duO This axiom holds.(8)This axiom fails.(9)c(du) = (cd)uThis axiom holds.O This axiom fails.(10) 1(u) = uO This axiom holds.O This axiom fails.STEP 2: Use your results from Step 1 to decide whether V is a vector space.O V is a vector space.O Vis not a vector space. Let V be the set of all positive real numbers. Determine whether V is a vector space with the following operations.х+у 3D хуAdditionCX = xCScalar multiplicationIf 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.u + v is in V.O This axiom holds.(1)This axiom fails.(2)u + v = v + uThis axiom holds.This axiom fails.u + (v + w) = (u + v) + wO This axiom holds.(3)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.

Given: V : set of all positive real numbers. And operations on V are defined as below: Addition: x…

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.

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.

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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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

(6)
cu is in V.
This axiom holds.
O This axiom fails.
(7)
c(u + v) = cu + cv
This axiom holds.
O This axiom fails.
(c + d)u = cu + du
O This axiom holds.
(8)
This axiom fails.
(9)
c(du) = (cd)u
This axiom holds.
O This axiom fails.
(10) 1(u) = u
O This axiom holds.
O This axiom fails.
STEP 2: Use your results from Step 1 to decide whether V is a vector space.
O V is a vector space.
O Vis not a vector space.

Let V be the set of all positive real numbers. Determine whether V is a vector space with the following operations.
х+у 3D ху
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.
u + v is in V.
O This axiom holds.
(1)
This axiom fails.
(2)
u + v = v + u
This axiom holds.
This axiom fails.
u + (v + w) = (u + v) + w
O This axiom holds.
(3)
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.

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