Linear Algebra

LINEAR ALGEBRA
 
EXPLORE A COMPREHENSIVE INTRODUCTORY TEXT IN LINEAR ALGEBRA WITH COMPELLING SUPPLEMENTARY MATERIALS, INCLUDING A COMPANION WEBSITE AND SOLUTIONS MANUALS
 
Linear Algebra delivers a fulsome exploration of the central concepts in linear algebra, including multidimensional spaces, linear transformations, matrices, matrix algebra, determinants, vector spaces, subspaces, linear independence, basis, inner products, and eigenvectors. While the text provides challenging problems that engage readers in the mathematical theory of linear algebra, it is written in an accessible and simple-to-grasp fashion appropriate for junior undergraduate students.
 
An emphasis on logic, set theory, and functions exists throughout the book, and these topics are introduced early to provide students with a foundation from which to attack the rest of the material in the text. Linear Algebra includes accompanying material in the form of a companion website that features solutions manuals for students and instructors. Finally, the concluding chapter in the book includes discussions of advanced topics like generalized eigenvectors, Schur's Lemma, Jordan canonical form, and quadratic forms. Readers will also benefit from the inclusion of:
* A thorough introduction to logic and set theory, as well as descriptions of functions and linear transformations
* An exploration of Euclidean spaces and linear transformations between Euclidean spaces, including vectors, vector algebra, orthogonality, the standard matrix, Gauss-Jordan elimination, inverses, and determinants
* Discussions of abstract vector spaces, including subspaces, linear independence, dimension, and change of basis
* A treatment on defining geometries on vector spaces, including the Gram-Schmidt process
 
Perfect for undergraduate students taking their first course in the subject matter, Linear Algebra will also earn a place in the libraries of researchers in computer science or statistics seeking an accessible and practical foundation in linear algebra.

List of contents

Preface xi
 
Acknowledgments xv
 
1 Logic and Set Theory 1
 
1.1 Statements 1
 
Connectives 2
 
Logical Equivalence 3
 
1.2 Sets and Quantification 7
 
Universal Quantification 8
 
Existential Quantification 9
 
Negating Quantification 10
 
Set-Builder Notation 12
 
Set Operations 13
 
Families of Sets 14
 
1.3 Sets and Proofs 18
 
Direct Proof 20
 
Subsets 22
 
Set Equality 23
 
Indirect Proof 24
 
Mathematical Induction 25
 
1.4 Functions 30
 
Injections 33
 
Surjections 35
 
Bijections and Inverses 37
 
Images and Inverse Images 40
 
Operations 41
 
2 Euclidean Space 49
 
2.1 Vectors 49
 
Vector Operations 51
 
Distance and Length 57
 
Lines and Planes 64
 
2.2 Dot Product 74
 
Lines and Planes 77
 
Orthogonal Projection 82
 
2.3 Cross Product 88
 
Properties 91
 
Areas and Volumes 93
 
3 Transformations and Matrices 99
 
3.1 Linear Transformations 99
 
Properties 103
 
Matrices 106
 
3.2 Matrix Algebra 116
 
Addition, Subtraction, and Scalar Multiplication 116
 
Properties 119
 
Multiplication 122
 
Identity Matrix 129
 
Distributive Law 132
 
Matrices and Polynomials 132
 
3.3 Linear Operators 137
 
Reflections 137
 
Rotations 142
 
Isometries 147
 
Contractions, Dilations, and Shears 150
 
3.4 Injections and Surjections 155
 
Kernel 155
 
Range 158
 
3.5 Gauss-Jordan Elimination 162
 
Elementary Row Operations 164
 
Square Matrices 167
 
Nonsquare Matrices 171
 
Gaussian Elimination 177
 
4 Invertibility 183
 
4.1 Invertible Matrices 183
 
Elementary Matrices 186
 
Finding the Inverse of a Matrix 192
 
Systems of Linear Equations 194
 
4.2 Determinants 198
 
Multiplying a Row by a Scalar 203
 
Adding a Multiple of a Row to Another Row 205
 
Switching Rows 210
 
4.3 Inverses and Determinants 215
 
Uniqueness of the Determinant 216
 
Equivalents to Invertibility 220
 
Products 222
 
4.4 Applications 227
 
The Classical Adjoint 228
 
Symmetric and Orthogonal Matrices 229
 
Cramer's Rule 234
 
LU Factorization 236
 
Area and Volume 238
 
5 Abstract Vectors 245
 
5.1 Vector Spaces 245
 
Examples of Vector Spaces 247
 
Linear Transformations 253
 
5.2 Subspaces 259
 
Examples of Subspaces 260
 
Properties 261
 
Spanning Sets 264
 
Kernel and Range 266
 
5.3 Linear Independence 272
 
Euclidean Examples 274
 
Abstract Vector Space Examples 276
 
5.4 Basis and Dimension 281
 
Basis 281
 
Zorn's Lemma 285
 
Dimension 287
 
Expansions and Reductions 290
 
5.5 Rank and Nullity 296
 
Rank-Nullity Theorem 297
 
Fundamental Subspaces 302
 
Rank and Nullity of a Matrix 304
 
5.6 Isomorphism 310
 
Coordinates 315
 
Change of Basis 320
 
Matrix of a Linear Transformation 324
 
6 Inner Product Spaces 335
 
6.1 Inner Products 335
 
Norms 341
 
Metrics 342
 
Angles 344
 
Orthogonal Projection 347
 
6.2 Orthonormal Bases 352
 
Orthogonal Complem

About the author

MICHAEL L. O'LEARY, is Professor of Mathematics at College of DuPage in Glen Ellyn, Illinois. He received his doctoral degree in mathematics from the University of California, Irvine in 1994 and is the author of A First Course in Mathematical Logic and Set Theory and Revolutions of Geometry, both published by Wiley.

Summary

LINEAR ALGEBRA

EXPLORE A COMPREHENSIVE INTRODUCTORY TEXT IN LINEAR ALGEBRA WITH COMPELLING SUPPLEMENTARY MATERIALS, INCLUDING A COMPANION WEBSITE AND SOLUTIONS MANUALS

Linear Algebra delivers a fulsome exploration of the central concepts in linear algebra, including multidimensional spaces, linear transformations, matrices, matrix algebra, determinants, vector spaces, subspaces, linear independence, basis, inner products, and eigenvectors. While the text provides challenging problems that engage readers in the mathematical theory of linear algebra, it is written in an accessible and simple-to-grasp fashion appropriate for junior undergraduate students.

An emphasis on logic, set theory, and functions exists throughout the book, and these topics are introduced early to provide students with a foundation from which to attack the rest of the material in the text. Linear Algebra includes accompanying material in the form of a companion website that features solutions manuals for students and instructors. Finally, the concluding chapter in the book includes discussions of advanced topics like generalized eigenvectors, Schur's Lemma, Jordan canonical form, and quadratic forms. Readers will also benefit from the inclusion of:
* A thorough introduction to logic and set theory, as well as descriptions of functions and linear transformations
* An exploration of Euclidean spaces and linear transformations between Euclidean spaces, including vectors, vector algebra, orthogonality, the standard matrix, Gauss-Jordan elimination, inverses, and determinants
* Discussions of abstract vector spaces, including subspaces, linear independence, dimension, and change of basis
* A treatment on defining geometries on vector spaces, including the Gram-Schmidt process

Perfect for undergraduate students taking their first course in the subject matter, Linear Algebra will also earn a place in the libraries of researchers in computer science or statistics seeking an accessible and practical foundation in linear algebra.

Product details

• Authors: Michael L O'Leary, Michael L. O'Leary, ML O'Leary, Michael L. O''leary, Michael L. (College of Dupage) O''leary, ML O''leary, O'Leary Michael L.
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 30.06.2021
• EAN: 9781119437444
• ISBN: 978-1-119-43744-4
• No. of pages: 464
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; Mathematik, Geometrie, Lineare Algebra, Mathematics, Linear Algebra, Geometrie u. Topologie, Logik u. Grundlagen der Mathematik, Geometry & Topology, Logic & Foundations