Author Profile

Chris McMullen a native of Manhattan and received dual majors in mathematics and psychology from the City College of New York. His unusual combination of academic interests led him toward a Master of Arts in mathematics from the University of Miami and a doctorate in behavioral sciences from Nova University. He is most energized by teaching mathematics and has taught a variety of mathematics courses at Miami-Dade College for nearly 30 years. He has received numerous teaching awards, including Innovator of the Year from the League for Innovations in the Community College, and was among the first group of recipients at Miami-Dade College for an endowed chair based on excellence in the classroom. He has written "Intermediate Algebra for College Students, Introductory Algebra for College Students, Essentials of Intermediate Algebra for College Students, Introductory and Intermediate Algebra for College Students, Essentials of Introductory and Intermediate Algebra for College Students, Algebra for College Students, Thinking Mathematically, College Algebra, Algebra and Trigonometry," and "Precalculus.

About The Book

Designed for the person who needs to learn Algebra as a prerequisite for further study or for a refresher course before moving on, the book covers all of the basic Algebra concepts such as variables, equations, Quadratic Equations, factoring algebraic Expressions, Exponents, roots, Radicals, and more. It includes numerous step by step examples and practice exercises that help the reader to understand the topics in a "self-study" Format, designed for those who are uncomfortable with Mathematics. With its use of multiple variables, functions, and formulas algebra can be confusing and overwhelming to learn and easy to forget. Perfect for students who need to review or reference critical concepts, Algebra Essentials provides content focused on key topics only, with discrete explanations of critical concepts taught in a typical Algebra course, from functions and FOILs to quadratic and linear equations. This book is also a perfect reference for parents who need to review critical algebra concepts. The book teaches the concepts and techniques of basic algebra with a focus on explaining definitions and theorems, and creating proofs. The theory is supported by numerous examples and plenty of worked-out problems. Its strict logical organization has been designed to help the reader to develop confidence in the subject. By introducing various interesting applications of algebra the book also aims at creating a broad and solid foundation for the study of advanced mathematics. The contents covered in the book are equivalence relations, functions, cardinality, congruence-modulo, mathematical induction and De Moivre's theorem. Further, some basic topics of linear algebra like vectors and matrices, linear equations, Gauss elimination, subspace and its dimension, rank-nullity theorem, linear transformations and their relations to matrices, and eigenvalues and eigenvectors are also covered. Since practice makes the man perfect, there are a good number of problems that stretch the thinking power of the learner. The problems are graded from easy to those involving higher order thinking. By its virtue the book inculcates that mathematical maturity which students need in their current and future courses to grow up into mathematicians of substance.

Table of Contents

Preface............................................................................................................. v
Chapter 1. Introduction to Algebra.................................................................. 1
’ Fundamentals of Algebra?...................................................................... 3
’ History and Evolution of Algebra..........................................................14
’ Variables, Constants, and Algebraic Expressions................................ 18
’ Operations on Algebraic Expressions.................................................. 23
Chapter 2. Linear Algebraic Concepts........................................................... 30
’ Solving Linear Equations in One Variable.............................................31
’ Graphing Linear Equations in Two Variables........................................ 35
’ Systems of Linear Equations................................................................39
’ Relation to Nonhomogeneous Systems..............................................49
’ Linear Inequalities and Their Graphs....................................................49
Chapter 3. Polynomials and Polynomial Functions................................... 56
’ Definition and Classification of Polynomials........................................ 57
’ Polynomial Operations.........................................................................65
’ The Division Algorithm for Polynomials...............................................67
’ Factorization of Polynomials.................................................................71
Chapter 4. Quadratic Equations and Expressions....................................... 78
’ Structure of Quadratic Equations........................................................80
’ The Quadratic Formula and Its Derivation...........................................97
’ Lagrange Resolvents........................................................................... 101
’ Nature of Roots and the Discriminant................................................103
Chapter 5. Functions and Their Properties................................................ 110
’ Definition of a Function........................................................................111
’ Domain and Range............................................................................... 117
’ Types of Functions...............................................................................120
Chapter 6. Complex Numbers...................................................................... 135
’ Introduction to Complex Numbers.....................................................136
Contents
’ Algebraic Properties of Complex Numbers........................................139
’ Geometric Representation (Argand Plane)........................................145
’ De Moivre’s Theorem...........................................................................147
Chapter 7. Sequences, Series, and Summations....................................... 151
’ Arithmetic Sequences and Series........................................................152
’ Geometric Sequences and Series........................................................157
’ Infinite Geometric Series.....................................................................163
’ Special Series (Square, Cube, etc.)......................................................165
Chapter 8. Matrices and Determinants....................................................... 170
’ Matrix Basics and Types....................................................................... 171
’ Matrix Addition, Multiplication, and Scalar Multiplication................177
’ Determinants and Their Properties.................................................... 180
’ Inverse of a Matrix.............................................................................. 189
Chapter 9. Vector Spaces and Subspaces.................................................... 196
’ Definition and Examples of Vector Spaces.........................................197
’ Subspaces and Spanning Sets............................................................ 205
’ Linear Independence...........................................................................215
’ Row and Column Spaces of Matrices..................................................218
Chapter 10. Group Theory Fundamentals.................................................. 225
’ Definition and Examples of Groups.................................................... 226
’ Group Axioms and Properties............................................................ 234
’ Subgroups and Cyclic Groups............................................................. 238
’ Permutation Groups and Symmetric Groups..................................... 244
Bibliography.............................................................................................. 251