Fundamentals of Linear Algebra

Preface

Advice to the Reader

1 Preliminaries

What is Linear Algebra?

Rudimentary Set Theory

Cartesian Products

Relations

Concept of a Function

Composite Functions

Fields of Scalars

Techniques for Proving Theorems

2 Matrix Algebra

Matrix Operations

Geometric Meaning of a Matrix Equation

Systems of Linear Equation

Inverse of a Matrix

The Equation Ax=b

Basic Applications

3 Vector Spaces

The Concept of a Vector Space

Subspaces

The Dimension of a Vector Space

Linear Independence

Application of Knowing dim (V)

Coordinates

Rank of a Matrix

4 Linear Maps

Linear Maps

Properties of Linear Maps

Matrix of a Linear Map

Matrix Algebra and Algebra of Linear Maps

Linear Functionals and Duality

Equivalence and Similarity

Application to Higher Order Differential Equations

5 Determinants

Motivation

Properties of Determinants

Existence and Uniqueness of Determinant

Computational Definition of Determinant

Evaluation of Determinants

Adjoint and Cramer's Rule

6 Diagonalization

Motivation

Eigenvalues and Eigenvectors

Cayley-Hamilton Theorem

7 Inner Product Spaces

Inner Product

Fourier Series

Orthogonal and Orthonormal Sets

Gram-Schmidt Process

Orthogonal Projections on Subspaces

8 Linear Algebra over Complex Numbers

Algebra of Complex Numbers

Diagonalization of Matrices with Complex Eigenvalues

Matrices over Complex Numbers

9 Orthonormal Diagonalization

Motivational Introduction

Matrix Representation of a Quadratic Form

Spectral Decompostion

Constrained Optimization-Extrema of Spectrum

Singular Value Decomposition (SVD)

10 Selected Applications of Linear Algebra

System of First Order Linear Differential Equations

Multivariable Calculus

Special Theory of Relativity

Cryptography

Solving Famous Problems from Greek Geometry

Answers to Selected Numberical Problems

Bibliography

Index