Complete the proof of Property 6 of the following theorem by supplying the justification for each step.Properties of Additive Identity and Additive InverseLet v be a vector in R", and let c be a scalar. Then the properties below are true.1. The additive identity is unique. That is, if v + u = v, then u = 0.2. The additive inverse of v is unique. That is, if v + u = 0, then u = -v.3. Ov = 04. c0 = 05. If cv = 0, then c = 0 or v = 0.6. -(-v) = vUse the properties of vector addition and scalar multiplication from the following theorem.Properties of Vector Addition and Scalar Multiplication in R"Let u, v, and w be vectors in R", and let c and d be scalars.1. u + v is a vector in R".Closure under addition2. u + v = v + uCommutative property of additionAssociative property of additionAdditive identity propertyAdditive inverse property3. (u + v) + w = u + (v + w)4. u + 0 = u5. u + (-u) = 06. cu is a vector in R"7. c(u + v) = cu + cv8. (c + d)u = cu + du9. c(du) = (cd)u10. 1(u) = uClosure under scalar multiplicationDistributive propertyDistributive propertyAssociative property of multiplicationMultiplicative identity propertyStepJustification-(-v) + (-v) = 0 and v + (-v) = 0---Select---(-v) + (-v) = v + (-v)transitive property of equality-(-v) + (-v) + v = v + (-v) + v-Select--- 6. -(-v) = vUse the properties of vector addition and scalar multiplication from the following theorem.Properties of Vector Addition and Scalar Multiplication in R"Let u, v, and w be vectors in R", and let c and d be scalars.1. u + v is a vector in R".Closure under addition2. u + v = v + uCommutative property of additionAssociative property of additionAdditive identity propertyAdditive inverse property3. (u + v) + w = u + (v + w)4. u + 0 = u5. u + (-u) = 06. cu is a vector in R".7. c(u + v) = cu + cvClosure under scalar multiplicationDistributive propertyDistributive property8. (c + d)u = cu + du9. c(du) = (cd)u10. 1(u) = uAssociative property of multiplicationMultiplicative identity propertyStepJustification-(-v) + (-v) = 0 and v + (-v) = 0-Select---Select--additive identity propertyadditive inverse propertyassociative property of multiplicationadd v to both sides-(-v) + (-v) = v + (-v)-(-v) + (-v) + v = v + (-v) + v-(-v) + ((-v) + v) = v + ((-v) + v)commutative property of additionassociative property of additiondistributive propertymultiplicative identity property-(-v) + 0 = v + 0-(-v) = v

Answered: Image /qna-images/answer/0458a071-117e-4ea2-b5d2-dfee86a53edf.jpg

Step Justification -(-v) + (-v) = 0 and v + (-v) = 0 ---Select--- --Select-- -(-v) + (-v) = v + (-v) additive identity property additive inverse property associative property of multiplication add v to both sides -(-v) + (-v) + v = v + (-v) + v -(-v) + ((-v) + v) = v + ((-v) + v) || commutative property of addition associative property of addition distributive property multiplicative identity property -(-v) + 0 = v + 0 -(-v) = v

Step Justification -(-v) + (-v) = 0 and v + (-v) = 0 ---Select--- --Select-- -(-v) + (-v) = v + (-v) additive identity property additive inverse property associative property of multiplication add v to both sides -(-v) + (-v) + v = v + (-v) + v -(-v) + ((-v) + v) = v + ((-v) + v) || commutative property of addition associative property of addition distributive property multiplicative identity property -(-v) + 0 = v + 0 -(-v) = v

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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