Definition 14.1: A vector space is a set V of objects, called vectors, on which two operations called addition and scalar multiplication have been defined satisfying the following properties. If u. v. w are in V and if a. 8 € R are scalars: (1) The sum u + v is in V. (closure under addition) (2) u+v=v+u (addition is commutative) (3) (u+v) +w=u+(v+w) (addition is associativity) (4) There is a vector in V called the zero vector, denoted by 0. satisfying v + 0 = v. (5) For each v there is a vector -v in V such that v + (-v) = 0. Vector Spaces (6) The scalar multiple of v by a, denoted av, is in V. (closure under scalar multiplica- tion) (7) a(u+v) = au + av (8) (a + 3)v=av + 3v (9) a(sv) = (aß)v (10) lv = v
Definition 14.1: A vector space is a set V of objects, called vectors, on which two operations called addition and scalar multiplication have been defined satisfying the following properties. If u. v. w are in V and if a. 8 € R are scalars: (1) The sum u + v is in V. (closure under addition) (2) u+v=v+u (addition is commutative) (3) (u+v) +w=u+(v+w) (addition is associativity) (4) There is a vector in V called the zero vector, denoted by 0. satisfying v + 0 = v. (5) For each v there is a vector -v in V such that v + (-v) = 0. Vector Spaces (6) The scalar multiple of v by a, denoted av, is in V. (closure under scalar multiplica- tion) (7) a(u+v) = au + av (8) (a + 3)v=av + 3v (9) a(sv) = (aß)v (10) lv = 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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Plz solve question 14.5
By using given information
Transcribed Image Text:Definition 14.1: A vector space is a set V of objects, called vectors, on which two
operations called addition and scalar multiplication have been defined satisfying the
following properties. If u. v. w are in V and if a. ß ER are scalars:
(1) The sum u + v is in V. (closure under addition)
(2) u+v=v+u (addition is commutative)
(3) (u+v) +w=u+ (v+w) (addition is associativity)
(4) There is a vector in V called the zero vector, denoted by 0. satisfying v + 0 = v.
(5) For each v there is a vector -v in V such that v + (-v) = 0.
Vector Spaces
(6) The scalar multiple of v by a, denoted av, is in V. (closure under scalar multiplica-
tion)
(7) a(u+v) = au + av
(8) (a + 3)v=av + 3v
(9) a(sv) = (aß)v
(10) lv = v
![Example 14.5. Let V = Pn[t] be the set of all polynomials in the variablet and of degree
at most n:
P₁[t] = {ao+at+ a₂t² +. + ant" | ao, a₁..., an ER
ER}.
Is V a vector space?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e598bbf-2d35-4afb-a9bb-7e240f86d153%2F2c9f379c-90d4-416b-8e97-46b9533bb8d8%2Friob5u8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Example 14.5. Let V = Pn[t] be the set of all polynomials in the variablet and of degree
at most n:
P₁[t] = {ao+at+ a₂t² +. + ant" | ao, a₁..., an ER
ER}.
Is V a vector space?
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