Advanced linear algebra - Boca Raton CRC Press 2016 - xix, 327 p. - Textbooks in Mathematics .

Table of Contents:

1. Fields and Matrix Algebra

The Field Z3

The Field Axioms

Field Examples

Matrix Algebra over Different Fields

Exercises

2. Vector Spaces

Definition of a Vector Space

Vector Spaces of Functions

Subspaces and More Examples of Vector Spaces

Linear Independence, Span, and Basis

Coordinate Systems

Exercises

3. Linear Transformations

Definition of a Linear Transformation

Range and Kernel of Linear Transformations

Matrix Representations of Linear Maps

Exercises

4. The Jordan Canonical Form

The Cayley-Hamilton Theorem

Jordan Canonical Form for Nilpotent Matrices

An Intermezzo about Polynomials

The Jordan Canonical Form

The Minimal Polynomial

Commuting Matrices

Systems of Linear Differential Equations

Functions of Matrices

The Resolvent

Exercises

5. Inner Product and Normed Vector Spaces

Inner Products and Norms

Orthogonal and Orthonormal Sets and Bases

The Adjoint of a Linear Map

Unitary Matrices, QR, and Schur Triangularization

Normal and Hermitian Matrices

Singular Value Decomposition

Exercises

6. Constructing New Vector Spaces from Given Ones

The Cartesian Product

The Quotient Space

The Dual Space

Multilinear Maps and Functionals

The Tensor Product

Anti-Symmetric and Symmetric Tensors

Exercises

7. How to Use Linear Algebra

Matrices You Can't Write Down, but Would Still Like to Use

Algorithms Based on Matrix Vector Products

Why Use Matrices When Computing Roots of Polynomials?

How to Find Functions with Linear Algebra?

How to Deal with Incomplete Matrices

Solving Millennium Prize Problems with Linear Algebra

How Secure Is RSA Encryption?

Quantum Computation and Positive Maps

Exercises

How to Start Your Own Research Project

Answers to Exercises

Advanced Linear Algebra features a student-friendly approach to the theory of linear algebra. The author’s emphasis on vector spaces over general fields, with corresponding current applications, sets the book apart. He focuses on finite fields and complex numbers, and discusses matrix algebra over these fields. The text then proceeds to cover vector spaces in depth. Also discussed are standard topics in linear algebra including linear transformations, Jordan canonical form, inner product spaces, spectral theory, and, as supplementary topics, dual spaces, quotient spaces, and tensor products.

Written in clear and concise language, the text sticks to the development of linear algebra without excessively addressing applications. A unique chapter on "How to Use Linear Algebra" is offered after the theory is presented. In addition, students are given pointers on how to start a research project. The proofs are clear and complete and the exercises are well designed. In addition, full solutions are included for almost all exercises.

(https://www.crcpress.com/Advanced-Linear-Algebra/Woerdeman/9781498754033)

9781498754033

Algebras, Linear

Cayley algebra

512.5 / W6A2