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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

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.CR: Review Exercises
Problem 73CR
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Plz solve question 14.5

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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
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?
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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