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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 = x² 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 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 V is 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. x + y = xy Addition CX = x² 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 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 V is not a vector space.

Advanced Engineering Mathematics
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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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. 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. 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 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 V is 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. 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. 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. 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 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 V is not a vector space.
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