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Elementary Linear Algebra: A Matrix Approach
Lawrence E. SpenceIllinois State University
Arnold J. InselIllinois State University
Stephen H. FriedbergIllinois State University

ISBN-10: 0137167229
ISBN-13:  9780137167227

Publisher:  Pearson
Copyright:  2000
Format:  Cloth; 451 pp
Published:  09/07/1999
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Description

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.


Features

  • Matrix orientation.
    • Allows instructors to present concepts more concretely. Ex.___

  • 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. Ex.___

  • 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. Ex.___

  • A large variety of applications.
    • Shows students the breadth and power of linear algebra. Ex.___

  • 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. Ex.___

  • 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. Ex.___

  • 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 R3.
  • Chapter reviews.
  • Boxed statements of important results.
  • Numbered (200) and unnumbered examples.


Table of Contents



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 R3 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.



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