## Elementary Linear Algebra: A Matrix Approach, 1/e

Lawrence E. Spence
Arnold J. Insel
Stephen H. Friedberg, all of Illinois State University

Published September, 1999 by Prentice Hall Engineering/Science/Mathematics

Cloth
ISBN 0-13-716722-9

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Introductory Linear Algebra-Mathematics

Linear Algebra-Mechanical Engineering

For a sophomore-level course in Linear Algebra. Based on the recommendations of the LACSG, this introduction to linear algebra offers a matrix-oriented approach with more emphasis on problem solving and applications and less emphasis on abstraction than in a traditional course. Throughout the text, use of technology is encouraged. The focus is on matrix arithmetic, systems of linear equations, properties of Euclidean n-space, eigenvalues and eigenvectors, and orthogonality. Although matrix-oriented, the text provides a solid coverage of vector spaces.

Matrix orientation.

• Allows instructors to present concepts more concretely.
Emphasis on Euclidean n-space rather than abstract vector spaces.
• Builds student confidence by first introducing concepts in a familiar setting. Then when students are ready (mature), vector spaces are covered.
A large number and variety of exercises (2200 in all)—Over 400 true/false questions, 100 practice problems (with solutions), theoretical exercises, and exercises that require the use of technology.
• Gives students the opportunity to test their understanding of basic concepts, practice performing important computations, delve more deeply into theory, formulate conjectures, and practice using technology.
A large variety of applications.
• Shows students the breadth and power of linear algebra.
Early introduction of the span and linear independence.
• Gives students a solid foundation in these key concepts by introducing them in familiar contexts early and by developing familiarity with them through computations.
Interplay between matrices and linear transformations—Linear transformations are introduced in Ch. 2 and go hand-in hand with matrices.
• Provides motivation for several topics for which the linear transformation version is more natural than the matrix formulation.
Optional sections that can be omitted without loss of continuity—e.g., Applications of Systems of Linear Equations; Applications of Matrix Multiplication; The LU Decomposition of a Matrix; Applications of Eigenvalues; Singular Value Decomposition; Rotations of R^3.
Chapter reviews.
Boxed statements of important results.
Numbered (200) and unnumbered examples.

1. Matrices, Vectors, and Systems of Linear Equations.

Matrices and Vectors. Linear Combinations, Matrix-Vector Products, and Special Matrices. Systems of Linear Equations. Gaussian Elimination. Applications of Systems of Linear Equations. The Span of a Set Vectors. Linear Dependence and Independence. Chapter 1 Review.

2. Matrices and Linear Transformations.

Matrix Multiplication. Applications of Matrix Multiplication. Invertibility and Elementary Matrices. The Inverse of a Matrix. The LU Decomposition of a Matrix. Linear Transformations and Matrices. Composition and Invertibility of Linear Transformations. Chapter 2 Review.

3. Determinants.

Cofactor Expansion. Properties of Determinants. Chapter 3 Review.

4. Subspaces and Their Properties.

Subspaces. Basis and Dimension. The Dimension of Subspaces Associated with a Matrix. Coordinate Systems. Matrix Representations of Linear Operators. Chapter 4 Review.

5. Eigenvalues, Eigenvectors, and Diagonalization.

Eigenvalues and Eigenvectors. The Characteristic Polynomial. Diagonalization of Matrices. Diagonalization of Linear Operators. Applications of Eigenvalues. Chapter 5 Review.

6. Orthogonality.

The Geometry of Vectors. Orthonormal Vectors. Least-Squares Approximation and Orthogonal Projection Matrices. Orthogonal Matrices and Operators. Symmetric Matrices. Singular Value Decomposition. Rotations of R^3 and Computer Graphics. Chapter 6 Review.

7. Vector Spaces.

Vector Spaces and their Subspaces. Dimension and Isomorphism. Linear Tranformations and Matrix Representations. Inner Product Spaces. Chapter 7 Review.

Appendix: Complex Numbers.