Work through each of the axioms in Def 3.5 to show that the set of polynomials of degree 3 or less, R3[X] := {ao + a₁X + a₂X² + α3X³ : a₁ ≤ R}, forms a vector space over the scalars R.
Work through each of the axioms in Def 3.5 to show that the set of polynomials of degree 3 or less, R3[X] := {ao + a₁X + a₂X² + α3X³ : a₁ ≤ R}, forms a vector space over the scalars R.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
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Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 37E: Let V be the set of all positive real numbers. Determine whether V is a vector space with the...
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![Definition 3.5. A vector space V over R (or C) is a non-empty set V consisting of
‘vectors' ū, V, w ... € V and ‘scalars' c = R (or C) together with ‘vector addition’ū+v and
'scalar multiplication' √ → cʊ satisfying
(a.) + EV and cv € V
(b.) ū+v=v+ū
(c.) ū+ (v+w) = (ū+ v) + w
(d.) 30€ V s.t. v + 0 = v
(e.) 3-u V s.t. ū+(-u) = 0
(f.) 1ū=ū
(g.) c(ữ +ừ) = cu trữ
(h.) (c+ c)ucu + c'u
[closure]
[commutativity]
[associativity]
[zero vector]
[negative vector]
[scalar 1 € R)]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9cb3be5-21be-4136-97fb-dd245561f49a%2F11ba04e1-4695-42e3-bfca-0bb4ee7d6377%2F05jgl2_processed.png&w=3840&q=75)
Transcribed Image Text:Definition 3.5. A vector space V over R (or C) is a non-empty set V consisting of
‘vectors' ū, V, w ... € V and ‘scalars' c = R (or C) together with ‘vector addition’ū+v and
'scalar multiplication' √ → cʊ satisfying
(a.) + EV and cv € V
(b.) ū+v=v+ū
(c.) ū+ (v+w) = (ū+ v) + w
(d.) 30€ V s.t. v + 0 = v
(e.) 3-u V s.t. ū+(-u) = 0
(f.) 1ū=ū
(g.) c(ữ +ừ) = cu trữ
(h.) (c+ c)ucu + c'u
[closure]
[commutativity]
[associativity]
[zero vector]
[negative vector]
[scalar 1 € R)]
![Work through each of the axioms in Def 3.5 to show that the set of polynomials
of degree 3 or less,
R3[X] := {ao + a₁X + a₂X² + α3X³ : a₁ ≤ R},
forms a vector space over the scalars R.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9cb3be5-21be-4136-97fb-dd245561f49a%2F11ba04e1-4695-42e3-bfca-0bb4ee7d6377%2F2ffoiwt_processed.png&w=3840&q=75)
Transcribed Image Text:Work through each of the axioms in Def 3.5 to show that the set of polynomials
of degree 3 or less,
R3[X] := {ao + a₁X + a₂X² + α3X³ : a₁ ≤ R},
forms a vector space over the scalars R.
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