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In Exercises 1–16 a set of objects is given, together with operations of addition and scalar multiplication. Determine which sets are vector spaces under the given operations. For those that are not vector spaces, list all axioms that fail to hold. The set of all triples of real numbers (x, y, z) with the operations 1. (x, y, 2) + (x', y',z') = (x +x',y +y',z+z') and k(x, y,z) = (kx,y,z)

In Exercises 1–16 a set of objects is given, together with operations of addition and scalar multiplication. Determine which sets are vector spaces under the given operations. For those that are not vector spaces, list all axioms that fail to hold. The set of all triples of real numbers (x, y, z) with the operations 1. (x, y, 2) + (x', y',z') = (x +x',y +y',z+z') and k(x, y,z) = (kx,y,z)

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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In Exercises 1-16 a set of objects is given, together with operations of addition and scalar multiplication. Determine which sets are vector spaces under the given operations. For those that are not vector spaces, list all axioms that fail to hold. The set of all triples of real numbers (x, y, z) with the operations 1. (x, y, 2) + (x', y', z') = (x + x', y +y', z+2) k(x, y, z) = (kx, y, z)
Transcribed Image Text:In Exercises 1-16 a set of objects is given, together with operations of addition and scalar multiplication. Determine which sets are vector spaces under the given operations. For those that are not vector spaces, list all axioms that fail to hold. The set of all triples of real numbers (x, y, z) with the operations 1. (x, y, 2) + (x', y', z') = (x + x', y +y', z+2) k(x, y, z) = (kx, y, z)
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