2. Determine whether the following sets with the corresponding vector addi- tion and scalar multiplication form vector spaces. a) Let V={(₁,₂): ₁, ₂ R). For (a₁,₂), (b₁,b₂) EV and e € R, define: (a₁.a₂)+(b₁,b₂)(a1 +2b₁,02+36₂) and c(a₁.₂) (ca.co₂) b) Let V=((a.az): 01.02 R). For (@a₁,0₂) V and c R, define: c(0₁.0₂) = (ca₂.0)
2. Determine whether the following sets with the corresponding vector addi- tion and scalar multiplication form vector spaces. a) Let V={(₁,₂): ₁, ₂ R). For (a₁,₂), (b₁,b₂) EV and e € R, define: (a₁.a₂)+(b₁,b₂)(a1 +2b₁,02+36₂) and c(a₁.₂) (ca.co₂) b) Let V=((a.az): 01.02 R). For (@a₁,0₂) V and c R, define: c(0₁.0₂) = (ca₂.0)
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:2. Determine whether the following sets with the corresponding vector addi-
tion and scalar multiplication form vector spaces.
a) Let V={(₁,₂): ₁, ₂ R). For (a,,a2), (b₁,b₂) EV and e R,
define:
(a₁.ay)+(b₁,b₂)=(a₁ +2b₁,02+36₂) and c(a₁.az) (ca.co₂)
b) Let V=((a.az): 01.02 € R). For (1,0₂) € Vand c
c(a.az) = (ca₁,0)
Step 1: Definition of vector space
A mon-empty set V is said to form a vector
ever a field F if
stence
(1) there is a binary composition + on V, called
addition, satisfying the conditions
1. d+pe V for all diper
2.
R+B=p+d for all di ptv.
1.
α+ (1+x) = (x+1)+d for all *₁1₁*6V.
4. there exists an element in Vluch that
1.
O Solution
, define:
Step 2: Check the properties
x+D = α for all dev
5. for each af in V there exists an element no
in V such that
and (1) there is an external composition of F
with V, called 'multiplicationly the elements
of F' satisfying the conditions,
6.
cat v for all c&f, all dev.
7. c(da) = (cd) α for all lide F all dev
8. C(R+p) = cateß for all off all Afv
9. (c+d) α = ca+dx for all of V, code F
10₁ 1α = α, 1 being the identity element inf.
0,
the
Here
given V = {(9₁, 9₂): 9₁, 9₂ ER}.
for (0,0₂), (b₁,b₂) EV and CEIR, define:
(o₁, A₂)+ (b₁,b₂) = (a₁ +26₁, A₂+36₂)
and c(0₁, as) = (cay, ca₂).
het, (0, ₂), (b₁,b₂), (C₁, C₂) EV and CidER.
(ay, as) + (bist₂) = (04+26₁, A₂+at₂) EV
(ou, as) + (bi, b₂) = (a₁ +26, ₂+ab₂)
(b₁,b₂) + (ay, a₂) = (by+201, b₂+ 39₂),
So, (ar, n) + (b₁,b₂) + (t₁,b₂) + (a₁, 92.)
Exemple, Consider, (1,1), (2,2) EV.
(1,1)+(2₁2) = (1+4, 1+6) = (5,7)
(212) + (1₁1) = (2+2, 2+3) = (4.5).
..V is not a vector space with respect to
addition and scalar multiplication.
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