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by Jimmie Gilbert and Linda Gilbert

ISBN13: 978-0534405816

ISBN10: 0534405819

Edition: 2ND 04

Copyright: 2004

Publisher: Brooks/Cole Publishing Co.

Published: 2004

International: No

ISBN10: 0534405819

Edition: 2ND 04

Copyright: 2004

Publisher: Brooks/Cole Publishing Co.

Published: 2004

International: No

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.

**Benefits: **

- NEW! Through rewritten material and increased examples and exercises, the pace of the early chapters, especially the first, has been made slower to help students can gain confidence early.
- NEW! Four new sections appear in Chapters 1 -10, including one on LU decompositions.
- NEW! A new Chapter 11, "Numerical Methods," enhances the usefulness of the text for two courses.
- The unifying concept for the first five chapters is that of elementary operations. The concept provides the pivot for a concise and efficient development of the basic theory of vector spaces, linear transformations, matrix multiplication, and the fundamental equivalence relations on matrices.
- A rigorous treatment of determinants from the traditional viewpoint is presented in Chapter 6.
- In Chapters 7 - 10, the central theme is the change in the matrix representing a function when only certain types of basis changes are admitted.
- Numerous examples and exercises illustrate the theory. The exercises are of both theoretical and computational nature. Those of a theoretical nature amplify the treatment and provide experience in constructing deductive arguments, while those of a computational nature illustrate fundamental techniques.

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.

Jimmie Gilbert and Linda Gilbert

ISBN13: 978-0534405816ISBN10: 0534405819

Edition: 2ND 04

Copyright: 2004

Publisher: Brooks/Cole Publishing Co.

Published: 2004

International: No

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.

**Benefits: **

- NEW! Through rewritten material and increased examples and exercises, the pace of the early chapters, especially the first, has been made slower to help students can gain confidence early.
- NEW! Four new sections appear in Chapters 1 -10, including one on LU decompositions.
- NEW! A new Chapter 11, "Numerical Methods," enhances the usefulness of the text for two courses.
- The unifying concept for the first five chapters is that of elementary operations. The concept provides the pivot for a concise and efficient development of the basic theory of vector spaces, linear transformations, matrix multiplication, and the fundamental equivalence relations on matrices.
- A rigorous treatment of determinants from the traditional viewpoint is presented in Chapter 6.
- In Chapters 7 - 10, the central theme is the change in the matrix representing a function when only certain types of basis changes are admitted.
- Numerous examples and exercises illustrate the theory. The exercises are of both theoretical and computational nature. Those of a theoretical nature amplify the treatment and provide experience in constructing deductive arguments, while those of a computational nature illustrate fundamental techniques.

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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