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Linear Algebra with Applications

Gareth Williams
Jones and Bartlett Publishers
Publication Date: 
Number of Pages: 
[Reviewed by
Marion Cohen
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“There should be something here to interest most people!” “I could not resist including a discussion of the use of vector space theory to detect errors in codes.” “This is mathematics at its best!” The author of this book certainly doesn’t hesitate to reveal his enthusiasm for his subject. This sincere enthusiasm comes through almost as clearly as his explanations. That this is a very good linear algebra text is probably an understatement.

Ideas are developed, theorems set off in “true blue” boxes, and most misconceptions anticipated. The theorems are also well-organized, and where possible many theorems are rolled into one (in particular, when many equivalent statements are involved).

This book covers the main ideas of Linear Algebra — systems of linear equations, matrices, determinants, vector spaces, eigenvalues and eigenvectors, linear transformations, inner product spaces, numerical techniques, and linear programming — in that order. Each chapter also contains applications, and the applications are interesting — in some cases amazing — and as well presented as the theories. A quick partial run-through: Applications of linear equations include fitting a polynomial of degree n - 1 to n given datapoints and analyses of electrical network and traffic flow; applications of matrices include archaeology, economics, demography and genetics, and communications and sociology; applications of eigenvectors include analyzing long-term trends of population movements (I especially loved the application to solving difference equations); applications of linear transformations include drawing a fractal fern; and applications of inner product spaces include approximating functions by polynomials, Non-Euclidean geometry, and a “self-contained discussion of the special relativity model of space-time, which involves the realization that we need PSEUDO-inner product rather than “just-plain” inner-product — and he doesn’t miss the opportunity to point out the “life-lesson” concerning the need to be ready to invent new structures with new axioms. Applications of numerical techniques include the concept of ill-conditioning. (To be complete — Linear Programming is of course an “application” in itself.)

I loved the exercises! In too many texts, there aren’t any, or aren’t enough, problems which give the student practice in the basic concepts and computations. This book has plenty of those, and they’re “isolated” at the very beginning of each set of exercises at the end of each section. In fact, he wisely groups the exercises according to topic, as well as to how challenging or non-straight-forward. Some exercises ask for proofs; most are not difficult but interesting nonetheless; one in particular struck me: “Prove that every symmetric triangular matrix is a diagonal matrix.” It’s “trivial”, but it gives the student the opportunity to painlessly cement the meanings of “symmetric”, “triangular”, and “diagonal”. The “true-false” problems are also interesting and instructive, likewise the occasional problem of the form, “Is Rn a subspace of Cn?”

Another Godsend in this book is Appendix C, p. 523 — a graphing calculator manual, for the TI83 and the TI83+. It contains, in four visually pleasing pages, the keystrokes for rref and matrix operations. There is also, with each topic of operation on the calculator, a sample problem to practice on, along with the answer. Also, Appendix D is a MATLAB manual.

However, as in the very first section of the first chapter of the book, I believe that it could be done even more clearly. For example, for the very beginning student (or the “newbies” to the calculator), “enter the matrix” doesn’t quite cut it. Some students need to be told, first, what that entails. I believe that, for many students, things need to be spelled out very clearly, and that authors of textbooks should take care not to take too much for granted.

That said, I’ll re-emphasize that this is a delightful book. I’d love to teach a course from it.

 Marion Cohen is Professor of Mathematics at the University of the Sciences in Philadelphia.


Chapter 1 Systems of Linear Equations
1.1 Matrices and Systems of Linear Equations
1.2 Gauss-Jordan Elimination
1.3* Curve Fitting, Electrical Networks, and Traffic Flow
Chapter 1 Review Exercises

Chapter 2 Matrices
2.1 Addition, Scalar Multiplication, and Multiplication of Matrices
2.2 Algebraic Properties of Matrix Operations
2.3 Symmetric Matrices and Seriation in Archaeology
2.4 Inverse of a Matrix, Elementary Matrices, Color Models and Cryptography
2.5* The Leontief Input-Output Model in Economics
2.6* Markov Chains, Population Movements, and Genetics
2.7* A Communication Model and Group Relationships in Sociology
Chapter 2 Review Exercises

Chapter 3 Determinants
3.1 Introduction to Determinants
3.2 Properties of Determinants
3.3* Numerical Evaluation of a Determinant
3.4 Determinants, Matrix Inverses, and Systems of Linear Equations
Chapter 3 Review Exercises

Chapter 4 Vector Spaces
4.1 The Vector Space Rn
4.2 Dot Product, Norm, Angle and Distance
4.3 General Vector Spaces
4.4 Subspaces
4.5 Linear Combinations of Vectors
4.6 Linear Dependence and Independence
4.7 Basis and Dimension
4.8 Rank of a Matrix
4.9 Orthonormal Vectors and Projections in Rn
Chapter 4 Review Exercises

Chapter 5 Eigenvalues and Eigenvectors
5.1 Eigenvalues, Eigenvectors, and Eigenspaces
5.2* Demography and Weather Prediction
5.3 Diagonalization of Matrices
5.4* Quadratic Forms, Difference Equations, and Normal Modes
Chapter 5 Review Exercises

Chapter 6 Linear Transformations
6.1 Introduction to Linear Transformations
6.2 Matrix Transformations, Computer Graphics, and Fractals
6.3 Kernel and Range
6.4* Transformations and Systems of Linear Equations
6.5 Coordinate Vectors
6.6 Matrix Representations of Linear Transformations
Chapter 6 Review Exercises

Chapter 7 Inner Product Spaces
7.1 Inner Product Spaces
7.2* Non-Euclidean Geometry and Special Relativity
7.3* Approximation of Functions and Coding Theory
7.4* Least Squares Curves
Chapter 7 Review Exercises

Chapter 8 Numerical Techniques
8.1* Gaussian Elimination
8.2* The Method of LU Decomposition
8.3* Practical Difficulties in Solving Systems of Equations
8.4* Iterative Methods for Solving Systems of Linear Equations
8.5* Eigenvalues by Iteration and Connectivity of Networks
Chapter 8 Review Exercises

Chapter 9 Linear Programming
9.1* A Geometrical Introduction to Linear Programming
9.2* The Simplex Method
9.3* Geometrical Explanation of the Simplex Method
Chapter 9 Review Exercises

A Cross Product
B Equations of Lines in Three-Space
C Graphing Calculator Manual

Answers to Selected Exercises

Appendix C Graphing Calculator Manual
C1 Solving Systems of Linear Equations in Two Variables Graphically (Section 1.1)
C2 Row Operations (Sections 1.1, 1.2, 1.3)
C3 Addition and Scalar Multiplication of Matrices (Section 2.1)
C4 Multiplication of Matrices and Powers of Matrices (Section 2.1)
C5 Transpose of a Matrix (Section 2.3)
C6 Inverse of a Matrix (Section 2.4)
C7 Determinant of a Matrix (Section 3.1)

Appendix D MATLAB Manual
D1 Entering and Displaying a Matrix (Section 1.1)
D2 Solving Systems of Linear Equations (Sections 1.1, 1.2, 1.3)
D3 Matrix Operations, Submatrices, Rows and Columns (Sections 2.1, 2.2, 2.3)
D4 Computational Considerations (Section 2.2)
D5 Inverse of a Matrix (Section 2.4)
D6 Solving Systems of Equations using Matrix Inverse (Section 2.4)
D7 Cryptography (Section 2.4)
D8 Leontief I/O Model (Section 2.5)
D9 Markov Chains (Sections 2.6, 6.2)
D10 Digraphs (Section 2.7)
D11 Determinants (Sections 3.1, 3.2, 3.3)
D12 Cramer's Rule (Section 3.4)
D13 Dot Product, Norm, Angle, Distance, (Section 4.2)
D14 Linear Combinations, Linear Dependence, Basis (Sections 4.4 - 4.7)
D15 Projection, Gram - Schmidt Orthogonalization (Section 4.8)
D16 Eigenvalues and Eigenvectors (Sections 5.1 - 5.4)
D17 Transformations defined by Matrices (Sections 6.1, 6.2)
D18 Fractals (Section 6.2)
D19 Kernel and Range (Section 6.3)
D20 Inner Product, Non-Euclidean Geometry (Sections 7.1, 7.2)
D21 Space-Time Travel (Section 7.2)
D22 Pseudoinverse and Least Squares Curves (Section 7.4)
D23 LU Decomposition (Section 8.2)
D24 Condition Number of a Matrix (Section 8.3)
D25 Jacobi and Gauss-Seidel Iterative Methods (Section 8.4)
D26 The Simplex Method in Linear Programming (Section 9.2)
D27 Cross Product ( Appendix A )
D28 MATLAB Commands, Functions and M-files
D29 The Linear Algebra with Applications Toolbox M-Files