Linear Algebra For Dummies

An easy-to-follow guide to practical, real-world applications of linear algebra

Linear algebra is a branch of mathematics that uses matrices to solve systems of linear equations; it has applications in many disciplines, from sociology and game theory to computer programming, engineering, and business. Linear Algebra For Dummies maps to a typical, college-level linear algebra course, in which students first study matrices and matrix operations, then apply those fundamentals to abstract topics such as vector spaces, linear transformations, determinants, and, finally, eigenvalues and eigenvectors. It gives students theoretical and practical ways of approaching various types of problems, presenting the information in a way that allows readers to fully digest not just the how of solving linear algebraic problems, but also the why.

Introduction 1

Part I: Lining Up the Basics of Linear Algebra 7

Chapter 1: Putting a Name to Linear Algebra 9

Chapter 2: The Value of Involving Vectors 19

Chapter 3: Mastering Matrices and Matrix Algebra 41

Chapter 4: Getting Systematic with Systems of Equations 65

Part II: Relating Vectors and Linear Transformations 85

Chapter 5: Lining Up Linear Combinations 87

Chapter 6: Investigating the Matrix Equation Ax = b 105

Chapter 7: Homing In on Homogeneous Systems and Linear Independence 123

Chapter 8: Making Changes with Linear Transformations 147

Part III: Evaluating Determinants 173

Chapter 9: Keeping Things in Order with Permutations 175

Chapter 10: Evaluating Determinants 185

Chapter 11: Personalizing the Properties of Determinants 201

Chapter 12: Taking Advantage of Cramer's Rule 223

Part IV: Involving Vector Spaces 239

Chapter 13: Involving Vector Spaces 241

Chapter 14: Seeking Out Subspaces of Vector Spaces 255

Chapter 15: Scoring Big with Vector Space Bases 273

Chapter 16: Eyeing Eigenvalues and Eigenvectors 289

Part V: The Part of Tens 309

Chapter 17: Ten Real-World Applications Using Matrices 311

Chapter 18: Ten (Or So) Linear Algebra Processes You Can Do on Your Calculator 327

Chapter 19: Ten Mathematical Meanings of Greek Letters 339

Glossary 343

Index 351

Mary Jane Sterling is the author of numerous For Dummies books. She is a lecturer at Bradley University in Peoria, Illinois, where she has taught courses in algebra, calculus, and other mathematics topics for almost 30 years.

Learn to:

• Solve linear algebra equations in several ways
• Put data in order with matrices
• Determine values with determinants
• Work with eigenvalues and eigenvectors

Your hands-on guide to real-world applications of linear algebra

Does linear algebra leave you feeling lost? No worries —this easy-to-follow guide explains the how and the why of solving linear algebra problems in plain English. From matrices to vector spaces to linear transformations, you'll understand the key concepts and see how they relate to everything from genetics to nutrition to spotted owl extinction.

• Line up the basics — discover several different approaches to organizing numbers and equations, and solve systems of equations algebraically or with matrices
• Relate vectors and linear transformations — link vectors and matrices with linear combinations and seek solutions of homogeneous systems
• Evaluate determinants — see how to perform the determinant function on different sizes of matrices and take advantage of Cramer's rule
• Hone your skills with vector spaces — determine the properties of vector spaces and their subspaces and see linear transformation in action
• Tackle eigenvalues and eigenvectors — define and solve for eigenvalues and eigenvectors and understand how they interact with specific matrices

Open the book and find:

• Theoretical and practical ways of solving linear algebra problems
• Definitions of terms throughout and in the glossary
• New ways of looking at operations
• How linear algebra ties together vectors, matrices, determinants, and linear transformations
• Ten common mathematical representations of Greek letters
• Real-world applications of matrices and determinants