A First Course in Linear Algebra

Minking Eie, Shou-Te Chang

• Publisher: WSPC
• Year: 2016
• ISBN:
  9789813143135
• Topic: Matematica
• Subtopic: Algebra

A First Course in Linear Algebra is written by two experts from algebra who have more than 20 years of experience in algebra, linear algebra and number theory. It prepares students with no background in Linear Algebra. Students, after mastering the materials in this textbook, can already understand any Linear Algebra used in more advanced books and research papers in Mathematics or in other scientific disciplines.

This book provides a solid foundation for the theory dealing with finite dimensional vector spaces. It explains in details the relation between linear transformations and matrices. One may thus use different viewpoints to manipulate a matrix instead of a one-sided approach. Although most of the examples are for real and complex matrices, a vector space over a general field is briefly discussed. Several optional sections are devoted to applications to demonstrate the power of Linear Algebra.

Contents:

• Preface
• Vector Spaces
• Bases and Dimension
• Linear Transformations and Matrices
• Elementary Matrix Operations
• Diagonalization
• Canonical Forms
• Inner Product Spaces

Readership: Undergraduates who are interested in learning linear algebra and its applications.

CHAPTER 1

Vector Spaces

It is probably the single most elementary and useful mathematical tool. We feel that Linear Algebra and Topology are the two prerequisites for any discipline of mathematics, while Linear Algebra is a prerequisite for any discipline that uses any mathematics at all.

In an algebraic course, it always involves the study of “algebraic structures” and the “morphisms” between the structures. For Linear Algebra, the “algebraic structures” are vector spaces.

1.1 A few words on sets and logics

Before we start, we need to review the basic language of mathematics: sets. We will also dwell slightly on how to write a proof. Remember that the business of mathematicians is to prove theorems.

Sets. We will say a set is a collection of elements. If an element x is contained in a set A, we will write x ∈ A. If not, we write x ∉ A. If we say two sets A and B are the same (equal), we mean that they contain exactly the same elements and we write A = B.

A set can be described by an account of all its elements or it can be described by a collection of sentences (conditions). Note that the conditions have to be specific enough to verify whether an element is inside this set or not.

Logical statements. A logical statement such as the sentences describing the conditions of an element in a set or making up a definition, cannot tolerate ambiguity. It is essential to familiarize oneself with the logical operators and rules of syntax.

A sentence (condition) must be of the form “(S)ubject + (V)erb + (O)bject”. In mathematics, =, >, <, ⊂, ⊃, ∈ and ∉ are verbs, while +, −, ÷, ∪ and ∩ are conjunctives. For example, “a > b” is a simple sentence (condition), while “a + b” is not!

A logical statement consists of conditions and logical operators, put together under the rules of syntax. Let (A) and (B) be sentences (conditions). The most commonly seen logical statements are two following types:

• ‘If (A) then (B)’, ‘(B) if (A)...