1. Let z be a fixed element of R. For arbitrary elements x, y = R" and arbitrary scalar a € R, define vector addition and scalar multiplication, denoted by and respectively, as follows: xy=x+y-z, Show that (R", 0, 0) is a vector space. = and ax a(x − 2) + z. - Definition 2.3.5 Consider a set V over a field F with given definitions for addition (+) and scalar multiplication (-). V with + and is called a vector space over F if for all u, v, w = V and for all a, ß € F, the following ten properties hold: (P1) Closure Property for Addition u + v € V. (P2) Closure Property for Scalar Multiplication a⚫ v € V. (P3) Commutative Property for Addition u + v = v+u. (P4) Associative Property for Addition (u + v) + w = u+(v+w). (P5) Associative Property for Scalar Multiplication a (v) = (aẞ). v. (P6) Distributive Property of Scalar Multiplication Over Vector Addition a (u + v) = a⋅u+ α·υ. . (P7) Distributive Property of Scalar Multiplication Over Scalar Addition (a + b) v = a· v + 3 . v. (P8) Additive Identity Property V contains the additive identity, denoted 0 so that 0 + v = v+0=v for every v € V. (P9) Additive Inverse Property V contains additive inverses z so that for every v = V there is a z = V satisfying v+z = 0. (P10) Multiplicative Identity Property for Scalars The scalar set F has an identity element, denoted 1, for scalar multiplication that has the property 1. v = v for every v € V.
1. Let z be a fixed element of R. For arbitrary elements x, y = R" and arbitrary scalar a € R, define vector addition and scalar multiplication, denoted by and respectively, as follows: xy=x+y-z, Show that (R", 0, 0) is a vector space. = and ax a(x − 2) + z. - Definition 2.3.5 Consider a set V over a field F with given definitions for addition (+) and scalar multiplication (-). V with + and is called a vector space over F if for all u, v, w = V and for all a, ß € F, the following ten properties hold: (P1) Closure Property for Addition u + v € V. (P2) Closure Property for Scalar Multiplication a⚫ v € V. (P3) Commutative Property for Addition u + v = v+u. (P4) Associative Property for Addition (u + v) + w = u+(v+w). (P5) Associative Property for Scalar Multiplication a (v) = (aẞ). v. (P6) Distributive Property of Scalar Multiplication Over Vector Addition a (u + v) = a⋅u+ α·υ. . (P7) Distributive Property of Scalar Multiplication Over Scalar Addition (a + b) v = a· v + 3 . v. (P8) Additive Identity Property V contains the additive identity, denoted 0 so that 0 + v = v+0=v for every v € V. (P9) Additive Inverse Property V contains additive inverses z so that for every v = V there is a z = V satisfying v+z = 0. (P10) Multiplicative Identity Property for Scalars The scalar set F has an identity element, denoted 1, for scalar multiplication that has the property 1. v = v for every v € V.
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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Question
Transcribed Image Text:1.
Let z be a fixed element of R. For arbitrary elements x, y = R" and arbitrary scalar a € R, define
vector addition and scalar multiplication, denoted by and respectively, as follows:
xy=x+y-z,
Show that (R", 0, 0) is a vector space.
=
and ax a(x − 2) + z.
-
Transcribed Image Text:Definition 2.3.5
Consider a set V over a field F with given definitions for addition (+) and scalar multiplication
(-). V with + and is called a vector space over F if for all u, v, w = V and for all a, ß € F, the
following ten properties hold:
(P1) Closure Property for Addition u + v € V.
(P2) Closure Property for Scalar Multiplication a⚫ v € V.
(P3) Commutative Property for Addition u + v = v+u.
(P4) Associative Property for Addition (u + v) + w = u+(v+w).
(P5) Associative Property for Scalar Multiplication
a (v) = (aẞ). v.
(P6) Distributive Property of Scalar Multiplication Over Vector Addition a (u + v) = a⋅u+
α·υ.
.
(P7) Distributive Property of Scalar Multiplication Over Scalar Addition (a + b) v = a· v +
3 . v.
(P8) Additive Identity Property V contains the additive identity, denoted 0 so that 0 + v =
v+0=v for every v € V.
(P9) Additive Inverse Property V contains additive inverses z so that for every v = V there is a
z = V satisfying v+z = 0.
(P10) Multiplicative Identity Property for Scalars The scalar set F has an identity element, denoted
1, for scalar multiplication that has the property 1. v = v for every v € V.
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