A Primer on Linear Algebra

✍ Scribed by I. N. Herstein, David J. Winter

• Publisher: Macmillan Publishing Company
• Year: 1988
• Tongue: English
• Leaves: 585
• Series: Macmillan
• Category: Library

✦ Table of Contents

Title
Preface
Contents
Symbol List
1. The 2x2 matrices
1.1. Introduction
1.2. Definitions and operations
1.3. Some notation
1.4. Trace, transpose, and odds and ends
1.5. Determinants
1.6. Cramer's rule
1.7. Mappings
1.8. Matrices as mappings
1.9. The Cayley-Hamilton theorem
1.10. Complex numbers
1.11. M2(C)
1.12. Inner products
2. Systems of linear equations
2.1. Introduction
2.2. Equivalent systems
2.3. Elementary row operations, echelon matrices
2.4. Solving systems of linear equations
3. The nxn matrices
3.1. The opening
3.2. Matrices as operators
3.3. Trace
3.4. Transpose and Hermitian adjoint
3.5. Inner product spaces
3.6. Bases of F^(n)
3.7. Change of basis of F^(n)
3.8. Invertible matrices
3.9. Matrices and bases
3.10. Bases and inner products
4. More on nxn matrices
4.1. Subspaces
4.2. More on subspaces
4.3. Gram-Schmidt orthogonalization process
4.4. Rank and nullity
4.5. Characteristic roots
4.6. Hermitian matrices
4.7. Triangularizing matrices with complex entries
4.8. Exponentials
5. Determinants
5.1. Introduction
5.2. Properties of determinants: row operations
5.3. Properties of determinants: column operations
5.4. Cramer's rule
5.5. Properties of determinants: other expansions
5.6. Elementary matrices
5.7. The determinant of the product
5.8. The characteristic polynomial
5.9. The Cayley-Hamilton theorem
6. Rectangular matrices. More on systems of linear equations
6.1. Rectangular matrices
6.2. Linear transformations from F^(n) to F^(m)
6.3. The nullspace and column space of an mxn matrix
7. Abstract vector spaces
7.1. Introduction, definitions, and examples
7.2. Subspaces
7.3. Homomorphisms and isomorphisms
7.4. Isomorphisms from V to F^(n)
7.5. Linear independence in infinite-dimensional vector spaces
7.6. Inner product spaces
7.7. More on inner product spaces
8. Linear transformations
8.1. Introduction
8.2. Definitions, examples, and some preliminary results
8.3. Products of linear transformations
8.4. Linear transformations as matrices
8.5. Hermitian ideas
8.6. Linear transformations from one space to another
9. Applications
9.1. Fibonacci numbers
9.2. Equations of curves
9.3. Markov processes
9.4. Incidence models
9.5. Differential equations
10. Least squares methods
10.1. Approximate solutions of systems of linear equations
10.2. The approximate inverse of an mxn matrix
10.3. Solving a matrix equation using its normal equation
10.4. Finding functions that approximate data
11. Linear algorithms
11.1. Introduction
11.2. The LDU factorization of A
11.3. The row reduction algorithm and its inverse
11.4. Back and forward substitution. Solving Ax = y
11.5. Approximate inverse and projection algorithms
11.6. A computer program for finding exact and approximate solutions
Index