Introduction to Linear Algebra

Rita Fioresi, Marta Morigi

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Print ISBN: 9781032793337
• eBook ISBN: 9781000427905
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

Linear algebra provides the essential mathematical tools to tackle all the problems in Science. Introduction to Linear Algebra is primarily aimed at students in applied fields (e.g. Computer Science and Engineering), providing them with a concrete, rigorous approach to face and solve various types of problems for the applications of their interest. This book offers a straightforward introduction to linear algebra that requires a minimal mathematical background to read and engage with.

Features

• Presented in a brief, informative and engaging style
• Suitable for a wide broad range of undergraduates
• Contains many worked examples and exercises

CHAPTER 1

Introduction to Linear Systems

In this chapter, we discuss how to solve linear systems with real coefficients using a method known as Gaussian algorithm. Later on, we will also use this method to solve other questions; at the same time, we will interpret linear systems as special cases of a much deeper theory.

1.1 LINEAR SYSTEMS: FIRST EXAMPLES

A linear equation is an equation where the unknowns appear with degree 1, that is an equation of the form:

$$\begin{array}{l}{a}_1{x}_1+a_2{x}_2+\dots +a_n{x}_n=b,\hfill \end{array}$$ (1.1)

where $a_1,a_2,\dots ,a_n$ and b are assigned numbers and $x_1,x_2,\dots ,x_n$ are the unknowns. The numbers $a_1,\dots ,a_n$ are called coefficients of the linear equation, b is called known term. If $b=0$ the equation is said to be homogeneous. A solution of the equation (1.1) is a n-tuple of numbers $(s_1,s_2,\dots ,s_n)$ that gives an equality when put in place of the unknowns. For example $(3,-1,4)$ is a solution of the equation $2x_1+7x_2-x_3=-5$ because $2·3+7·(-1)-4=-5$.

A linear system of m equations in n unknowns $x_1,x_2,\dots ,x_n$ is a set of m linear equations in n unknowns $x_1,x_2,\dots ,x_n$ that must be simultaneously satisfied:

$$\begin{array}{l}\left\{\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n=b_1\hfill \\ a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n=b_2\hfill \\ ⋮\hfill \\ {\mathrm{a...}}_{}\hfill \end{array}\right. \hfill \end{array}$$

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• Preface
• CHAPTER 1 ▪ Introduction to Linear Systems
• CHAPTER 2 ▪ Vector Spaces
• CHAPTER 3 ▪ Linear Combination and Linear Independence
• CHAPTER 4 ▪ Basis and Dimension
• CHAPTER 5 ▪ Linear Transformations
• CHAPTER 6 ▪ Linear Systems
• CHAPTER 7 ▪ Determinant and Inverse
• CHAPTER 8 ▪ Change of Basis
• CHAPTER 9 ▪ Eigenvalues and Eigenvectors
• CHAPTER 10 ▪ Scalar Products
• CHAPTER 11 ▪ Spectral Theorem
• CHAPTER 12 ▪ Applications of Spectral Theorem and Quadratic Forms
• CHAPTER 13 ▪ Lines and Planes
• CHAPTER 14 ▪ Introduction to Modular Arithmetic
• APPENDIX A ▪ Complex Numbers
• APPENDIX B ▪ Solutions of some suggested exercises
• Bibliography
• Index