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, y + ´y) and k(x, y) = (2kx, 2ky).

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, y + ´y) and k(x, y) = (2kx, 2ky).

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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Question

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, y + ´y) and k(x, y) = (2kx, 2ky).

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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