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Let V = R. For u, v EV and a E R define vector addition by uvu+v+1 and scalar multiplication by au= au+a - 1. It can be shown that (V, , ) is a vector space over the scalar field R. Find the following: the sum: 18-6= the scalar multiple: 901- the zero vector: Ov the additive inverse of : Bx=

Let V = R. For u, v EV and a E R define vector addition by uvu+v+1 and scalar multiplication by au= au+a - 1. It can be shown that (V, , ) is a vector space over the scalar field R. Find the following: the sum: 18-6= the scalar multiple: 901- the zero vector: Ov the additive inverse of : Bx=

Advanced Engineering Mathematics
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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Let V = R. For u, v EV and a ER define vector addition by uv := u + v + 1 and scalar multiplication by au:= au+a - 1. It can be shown that (V, , ) is a vector space over the scalar field R. Find the following: the sum: 18-6= the scalar multiple: 91= the zero vector: Ov the additive inverse of
Transcribed Image Text:Let V = R. For u, v EV and a ER define vector addition by uv := u + v + 1 and scalar multiplication by au:= au+a - 1. It can be shown that (V, , ) is a vector space over the scalar field R. Find the following: the sum: 18-6= the scalar multiple: 91= the zero vector: Ov the additive inverse of
Expert Solution
Step 1: Finding the values

(a) Finding the sum

u⊞v=u+v+1⇒1⊞-6=1+-6+1⇒1⊞-6=-4

(b) Finding the scalar multiple

a⊡u=au+a-1⇒9⊡1=91+9-1⇒9⊡1=17

(c) Finding the zero vector (0V)

Let v' be the zero vector

∴u⊞v'=u⇒          u+v'+1=u⇒-u+u+v'+1=-u+u⇒                 v'+1=0⇒            v'+1-1=0-1⇒                         v'=-1

Hence 0V=-1

(d) Finding the additive inverse

Let u' be the additive inverse

∴u⊞u'=0⇒u+u'+1=0⇒             u'=1-u

Hence the additive inverse is 1-u

 

 

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