You are here

Linear Mathematics: A Practical Approach

Patricia Clark Kenschaft
Publisher: 
Dover Publications
Publication Date: 
2013
Number of Pages: 
392
Format: 
Paperback
Price: 
22.95
ISBN: 
9780486497198
Category: 
Textbook
[Reviewed by
Allen Stenger
, on
11/25/2013
]

This is a beginner-level text in linear algebra and its applications that provides a lot of handholding and is full of worked examples. The stated prerequisites are two years of high-school algebra, and I think this is accurate; the book is aimed very low. It is a 2013 unaltered reprint of the 1978 Worth Publishing edition.

Roughly the first 150 pages are an introduction to linear algebra, in the traditional sense of linear equations, matrices, and determinants. Most of the rest of the book deals with methods and applications of linear programming, including a chapter on game theory and a chapter on the transportation problem (for which it covers streamlined methods such as then northwest corner method and the stepping stone method). There’s also a chapter on probability and counting, which may seem out of place but is used to introduce Markov processes.

The main weakness of the book is that there is no coverage of computers. There’s quite a lot of drill, but of a sort we today would expect the student to do on a computer.

Bottom line: A good, traditional, and concrete treatment, although for a course that probably doesn’t exist any more.


Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

  • Chapter 1. MATRICES: BASIC SKILLS AND APPLICATIONS
  •  
    • 1.1 Definitions, Addition, Scalar Multiplication, and Notation
    • 1.2 Parts-Listing an Input-Output Matrices; Triangula, Diagonal, and Symmetric Matrices
    • 1.3 Matrix Multiplication and Vector Inner Products
    • 1.4 Input-Output Models and Compact Notation
    • 1.5 Identities and Inverses
    • *1.6 Using Inverses in Cryptography
    •  
  • Chapter 2. GAUSS-JORDAN ROW OPERATIONS
  •  
    • 2.1 Linear Equations with a Unique Solution
    • 2.2 Linear Equations with a Unique Solution (continued)
    • 2.3 Elementary Matrices
    • 2.4 Finding the Multiplicative Inverse of a Matrix
    • 2.5 Using Inverses in Leontief Models
    • *2.6 Parts-Listing Problem and Accounting Model
    •  
  • Chapter 3. SYSTEMS OF LINEAR EQUATIONS WITHOUT UNIQUE SOLUTIONS
  •  
    • 3.1 Recognizing Nonunique Solutions
    • 3.2 Finding Nonunique Solutions of a System of Linear Equations
    • *3.3 Analysis of Traffic Flow Networks
    • 3.4 Geometric Interpretations of Linear Equations
    • *3.5 Linear Independence and Dependence and Row Rank
    •  
  • *Chapter 4. DETERMINANTS
  •  
    • 4.1 Classical Expansion of Determinants
    • 4.2 Uses of Determinants
    • 4.3 The Gauss-Jordan Method Applied to Determinants
    •  
  • Chapter 5 INTRODUCTION TO LINEAR PROGRAMMING
  •  
    • 5.1 Graphing Linear Inequalities
    • 5.2 Setting Up Linear Programming and the Graphical Approach
    • 5.3 Tabular Solutions of Linear Programming Problems
    • 5.4 Minimum Problems
    • *5.5 The Classic Diet Problem
    •  
  • Chapter 6. THE SIMPLEX ALGORITHM
  •  
    • 6.1 Solving Standard Linear Programming Problems Using the Simplex Algorithm
    • 6.2 Why the Simplex Algorithm Works
    • *6.3 Linear Programming Problems That Are Not Standard
    • *6.4 A Model of Cleaning a River at Minimum Cost
    •  
  • *Chapter 7. DUAL PROBLEMS
  •  
    • 7.1 Definition of Dual Problems and Economic Interpretation
    • 7.2 Solving for the Independent Variables in a Dual Problem
    • 7.3 Dual Problem Proofs
    •  
  • *Chapter 8. THE TRANSPORTATION PROBLEM
  •  
    • 8.1 Northwest Corner Algorithm and Minimum Cell Algorithm
    • 8.2 The Stepping-Stone Algorithm
    • 8.3 Harder Stepping Stones
    • 8.4 The Assignment Problem
    •  
  • Chapter 9. PROBABILITY
  •  
    • 9.1 Basic Concepts
    • *9.2 Counting
    • 9.3 Conditional Probability
    • *9.4 Regular Markov Matrices
    •  
  • Chapter 10. GAME THEORY
  •  
    • 10.1 Expected Value
    • 10.2 Saddle Points and Mixed Strategies
    • 10.3 Games and Matrices
    • 10.4 Solving Matrix Games Using the Simplex Algorithm
    •  
  • APPENDIXES
  • 1 Signed Numbers
  • 2 Slopes and Graphs of Linear Equations
  • 3 Solving Two Simultaneous Equations in Two Unknowns
  • ANSWERS TO SAMPLE TESTS
  • GLOSSARY
  • INDEX

Dummy View - NOT TO BE DELETED