8. P {p(x) in P2: p(0)=0} 9. P {p(x) in P2: p" (0) 0} = 10. P = {p(x) in P₂: p(x) = p(-x) for all x}
8. P {p(x) in P2: p(0)=0} 9. P {p(x) in P2: p" (0) 0} = 10. P = {p(x) in P₂: p(x) = p(-x) for all x}
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:DEFINITION 1
A set of elements V is said to be a vector space over a scalar field S if an addition
operation is defined between any two elements of V and a scalar multiplication
operation is defined between any element of S and any vector in V. Moreover,
if u, v, and w are vectors in V, and if a and b are any two scalars, then these
10 properties must hold.
Closure properties:
(c1) u + vis a vector in V.
(c2) av is a vector in V.
Properties of addition:
(al) u+v=v+u.
(a2)
u + (v + w) = (u + v) + w.
(a3) There is a vector in V such that v + 0 = v for all v in V.
(a4) Given a vector v in V, there is a vector -v in V such that v + (-v) = 0.
Properties of scalar multiplication:
(m1) a(bv) = (ab)v.
(m2)
a(u+v) = au + av.
(m3) (a + b)v = av + bv.
(m4)
lv = v for all v in V.
Transcribed Image Text:In Exercises 6-11, the given set is a subset of a vector
space. Which of these subsets are also vector spaces
in their own right? To answer this question, determine
whether the subset satisfies the 10 properties of Defini-
tion 1. (Note: Because these sets are subsets of a vector
space, properties (al), (a2), (m1), (m2), (m3), and (m4)
are automatically satisfied.)
6. S = {v in R4: v₁ + v₁ = 0}
7. S = {v in R: V₁ + V4 = 1}
8. P = {p(x) in P₂: p(0) = 0}
9. P = {p(x) in P₂: p" (0) # 0}
10. P = {p(x) in P₂: p(x) = p(-x) for all x}
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