Let V be the set of all positive real numbers. Determine whether V is a vector space with the following opera x + y = xy 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. O This axiom holds. O This axiom fails. (2) u + v = v + u O This axiom holds. O This axiom fails. (3) u + (v + w) = (u + v) + w O This axiom holds. O This axiom fails. (4) V has a zero vector 0 such that for every u in V, u + 0 = u. O This axiom holds. O This axiom fails.
Let V be the set of all positive real numbers. Determine whether V is a vector space with the following opera x + y = xy 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. O This axiom holds. O This axiom fails. (2) u + v = v + u O This axiom holds. O This axiom fails. (3) u + (v + w) = (u + v) + w O This axiom holds. O This axiom fails. (4) V has a zero vector 0 such that for every u in V, u + 0 = u. O This axiom holds. O 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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3.
Transcribed Image Text:Let V be the set of all positive real numbers. Determine whether V is a vector space with the following operation
x + y = xy
Cx = x
Addition
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.
O This axiom holds.
O This axiom fails.
(2) u + v = v + u
O This axiom holds.
O This axiom fails.
(3) u + (v + w) = (u + v) + w
O This axiom holds.
O This axiom fails.
(4) V has a zero vector 0 such that for every u in V, u + 0 = u.
O 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.
O This axiom holds.
O This axiom fails.
(6) cu is in V.
O This axiom holds.
O This axiom fails.
(7) c(u + v) = cu + cv
O This axiom holds.
O This axiom fails.
(8) (c + d)u = cu + du
O This axiom holds.
O This axiom fails.
(9) c(du) = (cd)u
O This axiom holds.
O This axiom fails.
(10) 1(u) = u
Ở This axiom holds.
O This axiom fails.
STEP 2:Use your results from Step 1 to decide whether Vis a vector space.
O vis a vector space.
O vis not a vector space.
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