9. Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on u = (u1, U2) and v = (v1, v2) : %3D u +v = (), Uzv2), ku = (ku1, ku2) %3D (a) Compute u +v and ku for u = (1,5), v= (2, - 2), and k = 4. (b) Show that (0, 0) # 0 in V. (d) Show that Axiom 5 holds by producing an ordered pair such that u +(-u) = 0 for u has zero component. (e) Find two vector space axioms that fail to hold. (c) Define the zero vector. %3D ha
9. Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on u = (u1, U2) and v = (v1, v2) : %3D u +v = (), Uzv2), ku = (ku1, ku2) %3D (a) Compute u +v and ku for u = (1,5), v= (2, - 2), and k = 4. (b) Show that (0, 0) # 0 in V. (d) Show that Axiom 5 holds by producing an ordered pair such that u +(-u) = 0 for u has zero component. (e) Find two vector space axioms that fail to hold. (c) Define the zero vector. %3D ha
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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Transcribed Image Text:9. Let V be the set of all ordered pairs of real numbers, and consider the following addition
and scalar multiplication operations on u = (u1, Uz) and v = (v1, v2):
u + v= (ujv1, U2V2), ku = (ku1, ku2)
(a) Compute u + v and ku for u = (1,5), v= (2, – 2), and k = 4.
(b) Show that (0,0) 0 in V.
(d) Show that Axiom 5 holds by producing an ordered pair such that
u + (-u) =0 for u has zero component.
(e) Find two vector space axioms that fail to hold.
%3D
(c) Define the zero vector.
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