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Elementary Linear Algebra: Applications Version

Howard Anton and Chris Rorres
John Wiley
Publication Date: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Herbert E. Kasube
, on

This text is an expanded version of Elementary Linear Algebra (Ninth Edition). The "expansion" comes in Chapter 11, which includes 21 applications of linear algebra. The applications that are included are diverse and not all typically found in other linear algebra texts. Included are discussions of geometric linear programming, Markov chains, game theory, temperature distributions, computed tomography, fractals, genetics and others. Each applications lists prerequisites so an instructor can decide how and when to discuss that section. It would probably be unreasonable for an instructor to cover all of these applications, but there is certainly a wonderful variety from which to choose.

The first ten chapters constitute the typical material for an elementary linear algebra course. The exposition is clear and concise. The exercise sets are lengthy and include problems of varying difficulty. Instructors would have a wide selection.

The addition of the applications chapter makes this solid textbook even more versatile.


Herbert Kasube is Professor of Mathematics at Bradley University in Peoria, IL.






Chapter 1. Systems of Linear Equations and Matrices.

1.1 Introduction to Systems of Linear Equations.

1.2 Gaussian Elimination.

1.3 Matrices and Matrix Operations.

1.4 Inverses; Rules of Matrix Arithmetic.

1.5 Elementary Matrices and a Method for Finding A-1.

1.6 Further Results on Systems of Equations and Invertibility.

1.7 Diagonal, Triangular, and Symmetric Matrices.

Chapter 2. Determinants.

2.1 Determinants by Cofactor Expansion.

2.2 Evaluating Determinants by Row Reduction.

2.3 Properties of the Determinant Function.

2.4 A Combinatorial Approach to Determinants.

Chapter 3. Vectors in 2 Space and 3-Space.

3.1 Introduction to Vectors (Geometric).

3.2 Norm of a Vector; Vector Arithmetic.

3.3 Dot Product; Projections.

3.4 Cross Product.

3.5 Lines and Planes in 3-Space.

Chapter 4. Euclidean Vector Spaces.

4.1 Euclidean n-Space.

4.2 Linear Transformations from Rn to Rm.

4.3 Properties of Linear Transformations from Rn to Rm.

4.4 Linear Transformations and Polynomials.

Chapter 5. General Vector Spaces.

5.1 Real Vector Spaces.

5.2 Subspaces.

5.3 Linear Independence.

5.4 Basis and Dimension.

5.5 Row Space, Column Space, and Nullspace.

5.6 Rank and Nullity.

Chapter 6. Inner Product Spaces.

6.1 Inner Products.

6.2 Angle and Orthogonality in Inner Product Spaces.

6.3 Orthonormal Bases: Gram-Schmidt Prodcess; QR-Decomposition.

6.4 Best Approximation; Least Squares.

6.5 Change of Basis.

6.6 Orthogonal Matrices.

Chapter 7. Eigenvalues, Eigenvectors.

7.1 Eigenvalues and Eigenvectors.

7.2 Diagonalization.

7.3 Orthogonal Diagonalization.

Chapter 8. Linear Transformations.

8.1 General Linear Transformations.

8.2 Kernel and range.

8.3 Inverse Linear Transformations.

8.4 Matrices of General Linear Transformations.

8.5 Similarity.

8.6 Isomorphism.

Chapter 9. Additional topics.

9.1 Application to Differential Equations.

9.2 Geometry and Linear Operators on R2.

9.3 Least Squares Fitting to Data.

9.4 Approximation Problems; Fourier Series.

9.5 Quadratic Forms.

9.6 Diagonalizing Quadratic Forms; Conic Sections.

9.7 Quadric Surfaces.

9.8 Comparison of Procedures for Solving Linear Systems.

9.9 LU-Decompositions.

Chapter 10. Complex Vector Spaces.

10.1 Complex Numbers.

10.2 Division of Complex Numbers.

10.3 Polar Form of a Complex Number.

10.4 Complex Vector Spaces.

10.5 Complex Inner Product Spaces.

10.6 Unitary Normal, and Hermitian Matrices.

Chapter 11. Applications of Linear Algebra.

11.1 Constructing Curves and Surfaces through Specified Points.

11.2 Electrical Networks.

11.3 Geometric Linear Programming.

11.4 The Earliest Applications of Linear Algebra.

11.5 Cubic Spline Interpolation.

11.6 Markov Chains.

11.7 Graph Theory.

11.8 Games of Strategy.

11.9 Leontief Economic Models.

11.10 Forest Management.

11.11 Computer Graphics.

11.12 Equilibrium Temperature Distributions.

11.13 Computed Tomography.

11.14 Fractals.

11.15 Chaos.

11.16 Cryptography.

11.17 Genetics.

11.18 Age-Specific Population Growth.

11.19 Harvesting of Animal Populations.

11.20 A Least Squares Model for Human Hearing.

11.21 Warps and Morphs.

Answers to Exercises.