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
Author(s): Mary Jane Sterling
Series: For Dummies
Edition: 1
Linear Algebra for Dummies
About the Author
Dedication
Author’s Acknowledgments
Contents at a Glance
Table of Contents
Introduction
About This Book
Conventions Used in This Book
What You’re Not to Read
Foolish Assumptions
How This Book Is Organized
Icons Used in This Book
Where to Go from Here
Part I: Lining Up the Basics of Linear Algebra
Chapter 1: Putting a Name to Linear Algebra
Solving Systems of Equations in Every Which Way but Loose
Matchmaking by Arranging Data in Matrices
Valuating Vector Spaces
Determining Values with Determinants
Zeroing In on Eigenvalues and Eigenvectors
Chapter 2: The Value of Involving Vectors
Describing Vectors in the Plane
Defining the Algebraic and Geometric Properties of Vectors
Managing a Vector’s Magnitude
Chapter 3: Mastering Matrices and Matrix Algebra
Getting Down and Dirty with Matrix Basics
Putting Matrix Operations on the Schedule
Putting Labels to the Types of Matrices
Connecting It All with Matrix Algebra
Investigating the Inverse of a Matrix
Chapter 4: Getting Systematic with Systems of Equations
Investigating Solutions for Systems
Dealing with Inconsistent Systems and No Solution
Solving Systems Algebraically
Revisiting Systems of Equations Using Matrices
Part II: Relating Vectors and Linear Transformations
Chapter 5: Lining Up Linear Combinations
Defining Linear Combinations of Vectors
Getting Your Attention with Span
Chapter 6: Investigating the Matrix Equation AX=b
Working Through Matrix-Vector Products
Confirming the Existence of a Solution or Solutions
Chapter 7: Homing In on Homogeneous Systems and Linear Independence
Seeking Solutions of Homogeneous Systems
Delving Into Linear Independence
Connecting Everything to Basis
Chapter 8: Making Changes with Linear Transformations
Formulating Linear Transformations
Proposing Properties of Linear Transformations
Writing the Matrix of a Linear Transformation
Determining the Kernel and Range of a Linear Transformation
Part III: Evaluating Determinants
Chapter 9: Keeping Things in Order with Permutations
Computing and Investigating Permutations
Involving Inversions in the Counting
Chapter 10: Determining Values of Determinants
Evaluating the Determinants of 2 × 2 Matrices
Using Determinants with Area and Volume
Chapter 11: Personalizing the Properties of Determinants
Transposing and Inverting Determinants
Interchanging Rows or Columns
Zeroing In on Zero Determinants
Manipulating Matrices by Multiplying and Combining
Taking on Upper or Lower Triangular Matrices
Determinants of Matrix Products
Chapter 12: Taking Advantage of Cramer’s Rule
Inviting Inverses to the Party with Determined Determinants
Solving Systems Using Cramer’s Rule
Recognizing and Dealing with a Nonanswer
Making a Case for Calculators and Computer Programs
Part IV: Involving Vector Spaces
Chapter 13: Promoting the Properties of Vector Spaces
Delving into the Vector Space
Describing the Two Operations
Singling Out the Specifics of Vector Space Properties
Chapter 14: Seeking Out Subspaces of a Vector Space
Investigating Properties Associated with Subspaces
Finding a Spanning Set for a Vector Space
Defining and Using the Column Space
Connecting Null Space and Column Space
Chapter 15: Scoring Big with Vector Space Bases
Going Geometric with Vector Spaces
Creating Bases from Spanning Sets
Making the Right Moves with Orthogonal Bases
Writing the Same Vector after Changing Bases
Chapter 16: Eyeing Eigenvalues and Eigenvectors
Defining Eigenvalues and Eigenvectors
Solving for Eigenvalues and Eigenvectors
Circling Around Special Circumstances
Getting It Straight with Diagonalization
Part V: The Part of Tens
Chapter 17: Ten Real-World Applications Using Matrices
Controlling Traffic
Catching Up with Predator-Prey
Creating a Secret Message
Saving the Spotted Owl
Migrating Populations
Plotting Genetic Code
Distributing the Heat
Making Economical Plans
Playing Games with Matrices
Eating Right
Chapter 18: Ten (Or So) Linear Algebra Processes You Can Do on Your Calculator
Letting the Graph of Lines Solve a System of Equations
Making the Most of Matrices
Performing Row Operations
Raising to Powers and Finding Inverses
Determining the Results of a Markov Chain
Solving Systems Using A–1*B
Adjusting for a Particular Place Value
Chapter 19: Ten Mathematical Meanings of Greek Letters
Insisting That π Are Round
Determining the Difference with Δ
Summing It Up with Σ
Row, Row, Row Your Boat with ρ
Taking on Angles with θ
Looking for a Little Variation with ε
Taking a Moment with μ
Looking for Mary’s Little λ
Wearing Your ΦΒΚ Key
Coming to the End with ω
Glossary
Index
