**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**