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

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 48E
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part c pleasee i have the first 3 please do the rest asap PLEASEE

 

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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