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4. Matrix A R³×3 is said to be upper triangular if ai,j = 0 for all i, j = 1,2,3 whenever i > j. Prove whether or not the set of all upper triangular 3 × 3 matrices is a vector space, with vector addition and scalar multiplication defined element-wise. Hint: Given our choice of vector addition and scalar multiplication, it suffices to check vector space properties 1,2. 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.

4. Matrix A R³×3 is said to be upper triangular if ai,j = 0 for all i, j = 1,2,3 whenever i > j. Prove whether or not the set of all upper triangular 3 × 3 matrices is a vector space, with vector addition and scalar multiplication defined element-wise. Hint: Given our choice of vector addition and scalar multiplication, it suffices to check vector space properties 1,2. 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)
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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4. Matrix A R³×3 is said to be upper triangular if ai,j = 0 for all i, j = 1,2,3 whenever i > j. Prove whether or not the set of all upper triangular 3 × 3 matrices is a vector space, with vector addition and scalar multiplication defined element-wise. Hint: Given our choice of vector addition and scalar multiplication, it suffices to check vector space properties 1,2.
Transcribed Image Text:4. Matrix A R³×3 is said to be upper triangular if ai,j = 0 for all i, j = 1,2,3 whenever i > j. Prove whether or not the set of all upper triangular 3 × 3 matrices is a vector space, with vector addition and scalar multiplication defined element-wise. Hint: Given our choice of vector addition and scalar multiplication, it suffices to check vector space properties 1,2.
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.
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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