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Determine which sets are vector spaces under the given operations. For those that are not, list all axioms that fail to hold. The set of all pairs of real numbers (x, y) with the operations (x, y) + (´x, y´) = (x + ´x + 1, y + ´y + 1) and k(x, y) = (kx, ky).

Determine which sets are vector spaces under the given operations. For those that are not, list all axioms that fail to hold. The set of all pairs of real numbers (x, y) with the operations (x, y) + (´x, y´) = (x + ´x + 1, y + ´y + 1) and k(x, y) = (kx, ky).

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...
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Determine which sets are vector spaces
under the given operations. For those that are not, list all axioms that fail
to hold.

The set of all pairs of real numbers (x, y) with the operations (x, y) +
(´x, y´) = (x + ´x + 1, y + ´y + 1) and k(x, y) = (kx, ky). 

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