Consider the line V = {(t,t + 1) |t € R} in R. Let V be the real vector space with vector addition e defined by (x, x + 1) (y, y + 1) = (x + y, x + y + 1) for each (x, x + 1), (y, y + 1) e V and scalar multiplication O defined by a © (x, x+ 1) = (ax, ax + 1) for each (x, x +1) e V and a E R. (a) Verify the distributive property a O (u e v) = (a ©u) (a O v) when a = 3, u = (-2, –1), v= (3, 4). (b) Prove the associative property u e (v €w)= (u€v) ew for all u, v, w € V.
Consider the line V = {(t,t + 1) |t € R} in R. Let V be the real vector space with vector addition e defined by (x, x + 1) (y, y + 1) = (x + y, x + y + 1) for each (x, x + 1), (y, y + 1) e V and scalar multiplication O defined by a © (x, x+ 1) = (ax, ax + 1) for each (x, x +1) e V and a E R. (a) Verify the distributive property a O (u e v) = (a ©u) (a O v) when a = 3, u = (-2, –1), v= (3, 4). (b) Prove the associative property u e (v €w)= (u€v) ew for all u, v, w € V.
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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Q4
Transcribed Image Text:Consider the line V = {(t,t + 1)|t e R} in R. Let V be the real vector space with vector
addition e defined by
(x, x + 1) & (y, y + 1) = (x + y, x + y + 1) for each (x, x + 1), (y, y + 1) e V
and scalar multiplication O defined by
a © (x, x + 1) = (ax, ax + 1) for each (x, x + 1) e V and a E R.
(a) Verify the distributive property a © (u e v) = (a © u) & (a © v) when
a = 3, u= (-2, –1),
v = (3, 4).
(b) Prove the associative property u (v Ow) = (u ☺ v) w for all u, v, w € V.
(c) Prove that u (0, 1) = u for all u E V.
(d) For each u e V find its additive inverse –u € V such that u (-u) = 0, where 0 is the
zero vector in V. Justify your answer by proving u (-u) = 0.
%3D
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