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.
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.
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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Question
Transcribed Image Text:cu is in V.
O This axiom holds.
(6)
This axiom fails.
(7) c(u + v) = cu + cv
This axiom holds.
O This axiom fails.
(8)
(c + d)u = cu + du
This axiom holds.
O This axiom fails.
(9)
c(du) = (cd)u
This axiom holds.
This axiom fails.
(10)
1(u) = u
This axiom holds.
O This axiom fails.
STEP 2: Use your results from Step 1 to decide whether V is a vector space.
O Vis a vector space.
O Vis not a vector space.
Transcribed Image Text: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 = x°
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.
O This axiom fails.
(2)
u + v = v + u
This axiom holds.
O This axiom fails.
(3)
u + (v + w) = (u + v) + w
This axiom holds.
O This axiom fails.
(4)
V has a zero vector 0 such that for every u in V, u + 0 = u.
This axiom holds.
O 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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