Linear Algebra as an Introduction to Abstract Mathematics

Isaiah Lankham, Bruno Nachtergaele, Anne Schilling

• Publisher: WSPC
• Year: 2015
• ISBN:
  9789814723794
• Topic: Mathematics
• Subtopic: Mathematics Reference
• Index: Mathematics

This is an introductory textbook designed for undergraduate mathematics majors with an emphasis on abstraction and in particular, the concept of proofs in the setting of linear algebra. Typically such a student would have taken calculus, though the only prerequisite is suitable mathematical grounding. The purpose of this book is to bridge the gap between the more conceptual and computational oriented undergraduate classes to the more abstract oriented classes. The book begins with systems of linear equations and complex numbers, then relates these to the abstract notion of linear maps on finite-dimensional vector spaces, and covers diagonalization, eigenspaces, determinants, and the Spectral Theorem. Each chapter concludes with both proof-writing and computational exercises.

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This is an introductory textbook designed for undergraduate mathematics majors with an emphasis on abstraction and in particular, the concept of proofs in the setting of linear algebra. Typically such a student would have taken calculus, though the only prerequisite is suitable mathematical grounding. The purpose of this book is to bridge the gap between the more conceptual and computational oriented undergraduate classes to the more abstract oriented classes. The book begins with systems of linear equations and complex numbers, then relates these to the abstract notion of linear maps on finite-dimensional vector spaces, and covers diagonalization, eigenspaces, determinants, and the Spectral Theorem. Each chapter concludes with both proof-writing and computational exercises.

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Readership: Undergraduates in mathematics.

Chapter 1

What is Linear Algebra?

1.1Introduction

This book aims to bridge the gap between the mainly computation-oriented lower division undergraduate classes and the abstract mathematics encountered in more advanced mathematics courses. The goal of this book is threefold:

(1)You will learn Linear Algebra, which is one of the most widely used mathematical theories around. Linear Algebra finds applications in virtually every area of mathematics, including multivariate calculus, differential equations, and probability theory. It is also widely applied in fields like physics, chemistry, economics, psychology, and engineering. You are even relying on methods from Linear Algebra every time you use an internet search like Google, the Global Positioning System (GPS), or a cellphone.

(2)You will acquire computational skills to solve linear systems of equations, perform operations on matrices, calculate eigenvalues, and find determinants of matrices.

(3)In the setting of Linear Algebra, you will be introduced to abstraction.As the theory of Linear Algebra is developed, you will learn how to make and use definitions and how to write proofs.

The exercises for each Chapter are divided into more computation-oriented exercises and exercises that focus on proof-writing.

1.2What is Linear Algebra?

Linear Algebra is the branch of mathematics aimed at solving systems of linear equations with a finite number of unknowns. In particular, one would like to obtain answers to the following questions:

• Characterization of solutions: Are there solutions to a given system of linear equations? How many solutions are there?
• Finding solutions: How does the solution set look? What are the solutions?

Linear Algebra is a systematic theory regarding the solutions of systems of linear equations.

Example 1.2.1. Let us take the following system of two linear equations in the two unknowns x₁ and x₂:

This system has a unique solution for x₁, x₂ ∈ ℝ, namely

and

.

The solution can be found in several different ways. One approach is to first solve for one of the unknowns in one of the equations and then to substitute the result into the other equation. Here, for example, we might solve to obtain

x₁ = 1 + x₂

from the second equation. Then, substituting this in place of x₁ in the first equation, we have

2(1 + x₂) + x₂ = 0.

From this, x₂ = −2/3. Then, by further substitution,

Alternatively, we can take a more systematic approach in eliminating variables. Here, for example, we can subtract 2 times the second equation from the first equation in order to obtain 3x₂ = −2. It is then immediate that

and, by substituting this value for x₂ in the first equation, that

.

Example 1.2.2. Take the following system of two linear equations in the two unknowns x₁ and x₂:

We can eliminate variables by adding −2 times the first equation to the second equation, which results in 0 = −1. This is obviously a contradiction, and hence this system of equations has no solution.

Example 1.2.3. Let us take the following sys...