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Let V be the set {(x, y) | x, y ≤ R} with addition operation defined by (x1, Y₁) ⇒ (X2, Y2) = (x1 + x2, Y₁Y2) and scalar multiplication defined by a Ⓒ (x, y) = (x, 0). Show that V is not a vector space by determining which of the 10 vector space axioms are not true for V. Enter your answer as a comma separated list such as 3, 5, 6 if axioms numbered 3, 5 and 6 fail to be true.

Let V be the set {(x, y) | x, y ≤ R} with addition operation defined by (x1, Y₁) ⇒ (X2, Y2) = (x1 + x2, Y₁Y2) and scalar multiplication defined by a Ⓒ (x, y) = (x, 0). Show that V is not a vector space by determining which of the 10 vector space axioms are not true for V. Enter your answer as a comma separated list such as 3, 5, 6 if axioms numbered 3, 5 and 6 fail to be true.

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Additional Topics In Trigonometry
Section3.3: Vectors In The Plane
Problem 3ECP
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The question has two parts: a) I have to find which of the axioms are not true for V. b) Explain why each vector space axiom I chose in a) failed to be true.
1. (Closed under addition:) The sum of u and v, denoted u + v, is in V. 2. (Closed under scalar multiplication:) The scalar multiple of u by a, denoted au, is in V. 3. (Addition is commutative:) u + v = v + u. 4. (Addition is associative:) (u + v) +w=u+(v+w). 5. (A zero vector exists:) There exists a vector 0 in V such that u + 0 = u. 6. (Additive inverses exist:) For each u in V, there exists a v in V such that u + v = 0. (We write v=-u.) 7. (Scaling by 1 is the identity:) lu = U. 8. (Scalar multiplication is associative): a(Bu) = (aß)u. 9. (Scalar multiplication distributes over vector addition:) a (u + v) = au + av. 10. (Scalar addition is distributive:) (a + B)u = au + ßu.
Transcribed Image Text:1. (Closed under addition:) The sum of u and v, denoted u + v, is in V. 2. (Closed under scalar multiplication:) The scalar multiple of u by a, denoted au, is in V. 3. (Addition is commutative:) u + v = v + u. 4. (Addition is associative:) (u + v) +w=u+(v+w). 5. (A zero vector exists:) There exists a vector 0 in V such that u + 0 = u. 6. (Additive inverses exist:) For each u in V, there exists a v in V such that u + v = 0. (We write v=-u.) 7. (Scaling by 1 is the identity:) lu = U. 8. (Scalar multiplication is associative): a(Bu) = (aß)u. 9. (Scalar multiplication distributes over vector addition:) a (u + v) = au + av. 10. (Scalar addition is distributive:) (a + B)u = au + ßu.
Let V be the set {(x, y) | x, y ≤ R} with addition operation defined by (x1, Y₁) = (x2, Y2) = (x1 + x2, Y₁Y2) and scalar multiplication defined by a (x, y) = (x, 0). Show that V is not a vector space by determining which of the 10 vector space axioms are not true for V. Enter your answer as a comma separated list such as 3, 5, 6 if axioms numbered 3, 5 and 6 fail to be true. 0
Transcribed Image Text:Let V be the set {(x, y) | x, y ≤ R} with addition operation defined by (x1, Y₁) = (x2, Y2) = (x1 + x2, Y₁Y2) and scalar multiplication defined by a (x, y) = (x, 0). Show that V is not a vector space by determining which of the 10 vector space axioms are not true for V. Enter your answer as a comma separated list such as 3, 5, 6 if axioms numbered 3, 5 and 6 fail to be true. 0
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