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(c) Verify all 10 axioms to show that this structure defines a vector space over the real scalars

(c) Verify all 10 axioms to show that this structure defines a vector space over the real scalars

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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part c pleaseeee

 

Consider the following vector structure. Let the vectors be ordered pairs (a, b) with a, b real numbers and b > 0. So examples of vectors here would be (V8, 2), (-5, ), (0, 7) Define the following operations on these ordered pairs. Note: Let k be any scalar with the scalars for this space being all real numbers. (a, b) O (c, d) = (ad + bc, bd) ko (a, b) = (kabk-1,8*) (a) Calculate (-4, 5) O (3, }) (b) Calculate – o (8, 4) (c) Verify all 10 axioms to show that this structure defines a vector space over the real scalars Hints 1) Remember the zero vector in a vector space is not necessarily just made of zeroes. 2) Remember the zero vector must satisfy: 0 = 0 0 v 3) Remember that additive inverses must satisfy, -v = -1 © v
Transcribed Image Text:Consider the following vector structure. Let the vectors be ordered pairs (a, b) with a, b real numbers and b > 0. So examples of vectors here would be (V8, 2), (-5, ), (0, 7) Define the following operations on these ordered pairs. Note: Let k be any scalar with the scalars for this space being all real numbers. (a, b) O (c, d) = (ad + bc, bd) ko (a, b) = (kabk-1,8*) (a) Calculate (-4, 5) O (3, }) (b) Calculate – o (8, 4) (c) Verify all 10 axioms to show that this structure defines a vector space over the real scalars Hints 1) Remember the zero vector in a vector space is not necessarily just made of zeroes. 2) Remember the zero vector must satisfy: 0 = 0 0 v 3) Remember that additive inverses must satisfy, -v = -1 © v
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