 # Linear Mathematics: A Practical Approach

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## Overview

Designed to help students appreciate the beauty of abstract patterns and the thrill of modeling the "real" world, this versatile, time-tested, and widely used text requires only two years of high school algebra. Suitable for a traditional one-year course in linear algebra or a more streamlined single-semester course, it can also serve for courses in finite mathematics or mathematics in the contemporary world for liberal arts students.
Carefully chosen examples and exercises form the basis of this treatment, in which students solve problems related to biology (nesting habits of birds), sociology (rural-urban migration patterns), transportation (traffic flow), psychology (verifying claims of ESP), business (maximizing profits), and other fields. Topics include matrices, Gauss-Jordan row operations, systems of linear equations without unique solutions, determinants, linear programming, the simplex algorithm, dual problems, probability, and game theory. Each chapter features sample tests with answers. ## Product Details

ISBN-13: 9780486497198 Dover Publications 06/19/2013 Dover Books on Mathematics Series 414 6.10(w) x 9.20(h) x 0.90(d)

Chapter 1 Matrices: Basic Skills and Applications 1

1.1 Definitions, Addition, Scalar Multiplication, and Notation 1

1.2 Parts-Listing and Input-Output Matrices; Triangular, Diagonal, and Symmetric Matrices 8

1.3 Matrix Multiplication and Vector Inner Products 20

1.4 Input-Output Models and Compact Notation 30

1.5 Identities and Inverses 36

*1.6 Using Inverses in Cryptography 44

Chapter 2 Gauss-Jordan Row Operations 49

2.1 Linear Equations with a Unique Solution 49

2.2 Linear Equations with a Unique Solution (continued) 59

2.3 Elementary Matrices 66

2.4 Finding the Multiplicative Inverse of a Matrix 72

2.5 Using Inverses in Leontief Models 78

*2.6 Parts-Listing Problem and Accounting Model 84

Chapter 3 Systems of Linear Equations without Unique Solutions 90

3.1 Recognizing Nonunique Solutions 90

3.2 Finding Nonunique Solutions of a System of Linear Equations 98

*3.3 Analysis of Traffic Flow Networks 109

3.4 Geometric Interpretations of Linear Equations 114

*3.5 Linear Independence and Dependence and Row Rank 123

*Chapter 4 Determinants 131

4.1 Classical Expansion of Determinants 131

4.2 Uses of Determinants 139

4.3 The Gauss-Jordan Method Applied to Determinants 145

Chapter 5 Introduction to Linear Programming 151

5.1 Graphing Linear Inequalities 151

5.2 Setting Up Linear Programming and the Graphical Approach 160

5.3 Tabular Solutions of Linear Programming Problems 173

5.4 Minimum Problems 181

*5.5 The Classic Diet Problem 187

Chapter 6 The Simplex Algorithm 194

6.1 Solving Standard Linear Programming Problems Using the Simplex Algorithm 194

6.2 Why the Simplex Algorithm Works 203

*6.3 Linear Programming Problems That Are Not Standard 212

*6.4 A Model of Cleaning a River at Minimum Cost 225

*Chapter 7 Dual Problems 234

7.1 Definition of Dual Problems and Economic Interpretation 234

7.2 Solving for the Independent Variables in a Dual Problem 240

7.3 Dual Problem Proofs 245

*Chapter 8 The Transportation Problem 250

8.1 Northwest Corner Algorithm and Minimum Cell Algorithm 250

8.2 The Stepping-Stone Algorithm 265

8.3 Harder Stepping Stones 274

8.4 The Assignment Problem 280

Chapter 9 Probability 293

9.1 Basic Concepts 293

*9.2 Counting 306

9.3 Conditional Probability 315

*9.4 Regular Markov Matrices 325

Chapter 10 Game Theory 336

10.1 Expected Value 336

10.2 Saddle Points and Mixed Strategies 342

10.3 Games and Matrices 355

10.4 Solving Matrix Games Using the Simplex Algorithm 364

Appendixes 371

1 Signed Numbers 371

2 Slopes and Graphs of Linear Equations 373

3 Solving Two Simultaneous Equations in Two Unknowns 380