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.
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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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.
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.
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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