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Step Justification -(-v) + (-v) = 0 and v + (-v) = 0 --Select-- -(-v) + (-v) = v + (-v) transitive property of equality -(-v) + (-v) + v = v + (-v) + v |---Select-- -(-v) + ((-v) + v) = v + ((-v) + v) --Select-- -(-v) + 0 = v + 0 ---Select-- -(-v) = v ---Select---

Step Justification -(-v) + (-v) = 0 and v + (-v) = 0 --Select-- -(-v) + (-v) = v + (-v) transitive property of equality -(-v) + (-v) + v = v + (-v) + v |---Select-- -(-v) + ((-v) + v) = v + ((-v) + v) --Select-- -(-v) + 0 = v + 0 ---Select-- -(-v) = v ---Select---

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.1: Vector In R^n
Problem 61E: Illustrate properties 110 of Theorem 4.2 for u=(2,1,3,6), v=(1,4,0,1), w=(3,0,2,0), c=5, and d=2....
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Complete the proof of Property 6 of the following theorem by supplying the justification for each step. Properties of Additive Identity and Additive Inverse Let 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 = 0 4. co = 0 5. If cv = 0, then c = 0 or v = 0. 6. -(-v) = v Use 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". 2. u + v = v + u Closure under addition Commutative property of addition Associative property of addition Additive identity property Additive inverse property Closure under scalar multiplication Distributive property 3. (u + v) + w = u + (v + w) 4. u + 0 = u 5. u + (-u) = 0 6. cu is a vector in R". 7. c(u + v) = cu + cv 8. (c + d)u = cu + du 9. c(du) = (cd)u 10. 1(u) = u Distributive property Associative property of multiplication Multiplicative identity property Step Justification -(-v) + (-v) = 0 and v + (-v) = 0 |---Select--- -(-v) + (-v) = v + (-v) transitive property of equality -(-v) + (-v) + v = v + (-v) + v ---Select--- -(-v) + ((-v) + v) = v + ((-v) + v) |--Select-- -(-v) + 0 = v + 0 ---Select--- -(-v) = v ---Select---
Transcribed Image Text:Complete the proof of Property 6 of the following theorem by supplying the justification for each step. Properties of Additive Identity and Additive Inverse Let 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 = 0 4. co = 0 5. If cv = 0, then c = 0 or v = 0. 6. -(-v) = v Use 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". 2. u + v = v + u Closure under addition Commutative property of addition Associative property of addition Additive identity property Additive inverse property Closure under scalar multiplication Distributive property 3. (u + v) + w = u + (v + w) 4. u + 0 = u 5. u + (-u) = 0 6. cu is a vector in R". 7. c(u + v) = cu + cv 8. (c + d)u = cu + du 9. c(du) = (cd)u 10. 1(u) = u Distributive property Associative property of multiplication Multiplicative identity property Step Justification -(-v) + (-v) = 0 and v + (-v) = 0 |---Select--- -(-v) + (-v) = v + (-v) transitive property of equality -(-v) + (-v) + v = v + (-v) + v ---Select--- -(-v) + ((-v) + v) = v + ((-v) + v) |--Select-- -(-v) + 0 = v + 0 ---Select--- -(-v) = v ---Select---
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