# Fundamentals of Linear Algebra

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## Book Description

Fundamentals of Linear Algebra is like no other book on the subject. By following a natural and unified approach to the subject it has, in less than 250 pages, achieved a more complete coverage of the subject than books with more than twice as many pages. For example, the textbooks in use in the United States prove the existence of a basis only for finite dimensional vector spaces. This book proves it for any given vector space.

With his experience in algebraic geometry and commutative algebra, the author defines the dimension of a vector space as its Krull dimension. By doing so, most of the facts about bases when the dimension is finite, are trivial consequences of this definition. To name one, the replacement theorem is no longer needed. It becomes obvious that any two bases of a finite dimensional vector space contain the same number of vectors. Moreover, this definition of the dimension works equally well when the geometric objects are nonlinear.

Features:

- Presents theories and applications in an attempt to raise expectations and outcomes
- The subject of linear algebra is presented over arbitrary fields
- Includes many non-trivial examples which address real-world problems

About the Author:

**Dr. J.S. Chahal** is a professor of mathematics at Brigham Young University. He received his Ph.D. from Johns Hopkins University and after spending a couple of years at the University of Wisconsin as a post doc, he joined Brigham Young University as an assistant professor and has been there ever since. He specializes and has published a number of papers about number theory. For hobbies, he likes to travel and hike, the reason he accepted the position at Brigham Young University

## Table of Contents

**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**

## Author(s)

### Biography

**Dr. J.S. Chahal** is a professor of mathematics at Brigham Young University. He received his Ph.D. from Johns Hopkins University and after spending a couple of years at the University of Wisconsin as a post doc, he joined Brigham Young University as an assistant professor and has been there ever since. He specializes and has published a number of papers about number theory. For hobbies, he likes to travel and hike, the reason he accepted the position at Brigham Young University.