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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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.1: Vector Spaces And Subspaces
Problem 7EQ
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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.
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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