Linear Algebra and Matrix Theory / Edition 2

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Overview

Intended for a serious first course or a second course in linear algebra, this book carries students beyond eigenvalues and eigenvectors to the classification of bilinear forms, normal matrices, spectral decompositions, the Jordan form, and sequences and series of matrices. The authors present the material from a structural point of view: fundamental algebraic properties of the entities involved are emphasized. The approach is particularly important because the mathematical systems encountered in linear algebra furnish a wealth of examples for the structures studied in more advanced courses. By taking a straight and smooth path to the heart of linear algebra, students will be able to make the transition from the intuitive developments of courses at a lower level to the more abstract treatments encountered later.

Audience: Undergraduate math majors who have some background in linear algebra and would benefit from a strong foundation for more abstract treatments at a higher level. Also serves as text for introductory courses.

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Editorial Reviews

From the Publisher
"Jimmie and Linda Gilbert...present material at the appropriate level in a quick and concise manner....Some texts do not develop enough mathematical sophistication to enable the transition to junior- or senior-level linear algebra courses. The Gilbert text does!" "While the earlier edition of the Gilbert book was suitable for second and third year mathematics students, the level of writing was challenging for first year students. I was very pleased to find that the Gilberts have added a good number of meaningful examples and exercises at the appropriate level for such students. Moreover, in carefully doing so, they have not destroyed their crisp and efficient style of writing."

"I am...quite enthusiastic about the arrangement of topics which the Gilberts have chosen....The Gilberts Chapter 1 plays as a sort of overture, in which the main elements of vector space theory are laid out in a comfortable space, while due attention is paid to the role of computation. I think this type of start to a course would orient the students properly toward general vector space theory, rather than having them come upon the annoying complication of vector spaces after three or more weeks spent exclusively in computation and row reduction."

"The notation and terminology are clear and concise and the flow of topics and concepts is smooth."

"The material is well-written, it will be easy to teach from, and the students will find it easy to read and study from."

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Product Details

  • ISBN-13: 9780534405816
  • Publisher: Cengage Learning
  • Publication date: 2/16/2004
  • Edition description: REV
  • Edition number: 2
  • Pages: 552
  • Product dimensions: 7.50 (w) x 9.40 (h) x 0.90 (d)

Meet the Author

Jimmie Gilbert was Professor of Mathematics at the University of South Carolina, Upstate. He received his Ph.D from Auburn University with a specialty in Linear and Abstract Algebras. He authored the first edition of Elements of Modern Algebra in 1970, joined on subsequent editions by his wife and longtime co-author Linda Gilbert. Together they have published titles in College Algebra, Precalculus, College Algebra and Trigonometry, Trigonometry, Intermediate Algebra, and another Cengage Learning title, Linear Algebra and Matrix Theory, now in its second edition. He and Linda have 6 children and 8 grandchildren. In his leisure time Jimmie enjoyed the outdoors, fishing, and gardening.

Linda Gilbert received her Ph.D. from Louisiana Tech University with a specialty in Linear and Abstract Algebras. She has been writing textbooks since 1981 with her husband Jimmie Gilbert, including ELEMENTS OF MODERN ALGEBRA and LINEAR ALGEBRA and MATRIX THEORY (now in its second edition) with Cengage Learning, plus titles in College Algebra, Precalculus, College Algebra and Trigonometry, Trigonometry, and Intermediate Algebra.

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Table of Contents

1. REAL COORDINATE SPACES. The Vector Spaces Rn. Linear Independence. Subspaces of Rn. Spanning Sets. Geometric Interpretations of R² and R³. Bases and Dimension. 2. ELEMENTARY OPERATIONS ON VECTORS. Elementary Operations and Their Inverses. Elementary Operations and Linear Independence. Standard Bases for Subspaces. 3. MATRIX MULTIPLICATION. Matrices of Transition. Properties of Matrix Multiplication. Invertible Matrices. Column Operations and Column-Echelon Forms. Row Operations and Row-Echelon Forms. Row and Column Equivalence. Rank and Equivalence. LU Decompositions. 4. VECTOR SPACES, MATRICES, AND LINEAR EQUATIONS. Vector Spaces. Subspaces and Related Concepts. Isomorphisms of Vector Spaces. Standard Bases for Subspaces. Matrices over an Arbitrary Field. Systems of Linear Equations. More on Systems of Linear Equations. 5. LINEAR TRANSFORMATIONS. Linear Transformations. Linear Transformations and Matrices. Change of Basis. Composition of Linear Transformations. 6. DETERMINANTS. Permutations and Indices. The Definition of a Determinant. Cofactor Expansions. Elementary Operations and Cramer's Rule. Determinants and Matrix Multiplication. 7. EIGENVALUES AND EIGENVECTORS. Eigenvalues and Eigenvectors. Eigenspaces and Similarity. Representation by a Diagonal Matrix. 8. FUNCTIONS OF VECTORS. Linear Functionals. Real Quadratic Forms. Orthogonal Matrices. Reduction of Real Quadratic Forms. Classification of Real Quadratic Forms. Binlinear Forms. Symmetric Bilinear Forms. Hermitian Forms. 9. INNER PRODUCT SPACES. Inner Products. Norms and Distances. Orthonormal Bases. Orthogonal Complements. Isometrics. Normal Matrices. Normal Linear Operators. 10. SPECTRAL DECOMPOSITIONS. Projections and Direct Sums. Spectral Decompositions. Minimal Polynomials and Spectral Decompositions. Nilpotent Transformations. The Jordan Canonical Form. 11. NUMERICAL METHODS. Sequences and Series of Vectors. Sequences and Series of Matrices. The Standard Method of Iteration. Cimmino's Method. An Iterative Method for Determining Eigenvalues.

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