Linear Algebra with Applications

Roger Baker, Kenneth Kuttler

• Publisher: WSPC
• Year: 2014
• ISBN:
  9789814590556
• Topic: Mathématiques
• Subtopic: Théorie des nombres

This book gives a self- contained treatment of linear algebra with many of its most important applications. It is very unusual if not unique in being an elementary book which does not neglect arbitrary fields of scalars and the proofs of the theorems. It will be useful for beginning students and also as a reference for graduate students and others who need an easy to read explanation of the important theorems of this subject.

It presents a self- contained treatment of the algebraic treatment of linear differential equation which includes all proofs. It also contains many different proofs of the Cayley Hamilton theorem. Other applications include difference equations and Markov processes, the latter topic receiving a more thorough treatment than usual, including the theory of absorbing states. In addition it contains a complete introduction to the singular value decomposition and related topics like least squares and the pseudo-inverse.

Most major topics receive more than one discussion, one in the text and others being outlined in the exercises. The book also gives directions for using maple in performing many of the difficult algorithms.

Contents:

• Numbers, Vectors and Fields
• Matrices
• Row Operations
• Vector Spaces
• Linear Mappings
• Inner Product Spaces
• Similarity and Determinants
• Characteristic Polynomial and Eigenvalues of a Matrix
• Some Applications
• Unitary, Orthogonal, Hermitian and Symmetric Matrices
• The Singular Value Decomposition

Readership: Undergraduates in linear algebra.
Key Features:

• This book proves all the theorems including theorems about the determinant
• This book does not neglect algebraic considerations like general fields
• The applications of linear algebra are given a reasonably complete development

Chapter 1

Numbers, vectors and fields

Chapter summary

This chapter is on fundamental notation concerning sets and gives a brief introduction to fields, one of the ingredients which is important in the study of linear algebra. It contains a specific example of a vector space ℝ³ along with some of the geometric ideas concerning vector addition and the dot and cross product. It also contains a description of the most important examples of fields. These include the real numbers, complex numbers, and the field of residue classes.

1.1 Functions and sets

A set is a collection of things called elements of the set. For example, one speaks of the set of integers, whole numbers such as 1,2,−4, and so on. This set, whose existence will be assumed, is denoted by ℤ. The symbol ℝ will denote the set of real numbers which is usually thought of as points on the number line. Other sets could be the set of people in a family or the set of donuts in a display case at the store. Sometimes parentheses, { } specify a set by listing the things which are in the set between the parentheses. For example, the set of integers between −1 and 2, including these numbers, could be denoted as {−1, 0, 1, 2}. The notation signifying x is an element of a set S, is written as x ∈ S. We write x ∉ S for ‘x is not an element of S’. Thus, 1 ∈ {−1, 0, 1, 2, 3} and 7 ∉ {−1, 0, 1, 2, 3}. Here are some axioms about sets. Axioms are statements which are accepted, not proved.

(1)Two sets are equal if and only if they have the same elements.

(2)To every set A, and to every condition S(x) there corresponds a set B, whose elements are exactly those elements x of A for which S(x) holds.

(3)For every collection of sets there exists a set that contains all the elements that belong to at least one set of the given collection. This set is called the union of the sets.

Example 1.1. As an illustration of these axioms, {1, 2, 3} = {3, 2, 1} because they have the same elements. If you consider ℤ the set of integers, let S(x) be the condition that x is even. Then the set B consisting of all elements x of ℤ such that S(x) is true, specifies the even integers. We write this set as follows:

{x ∈ ℤ : S(x)}

Next let A = {1, 2, 4}, B = {2, 5, 4, 0}. Then the union of these two sets is the set {1, 2, 4, 5, 0}. We denote this union as A ∪ B.

The following is the definition of a function.

Definition 1.1. Let X, Y be nonempty sets. A function f is a rule which yields a unique y ∈ Y for a given x ∈ X. It is customary to write f (x) for this element of Y. It is also customary to write

f : X → Y

to indicate that f is a function defined on X which gives an element of Y for each x ∈ X.

Example 1.2. Let X = ℝ and f (x) = 2x.

The following is another general consideration.

Definition 1.2. Let f : X → Y where f is a function. Then f is said to be one-toone (injective), if whenever x₁ ≠ x₂, it follows that f (x₁) ≠ f (x₂). The function is said to be onto, (surjective), if whenever y ∈ Y, there exists an x ∈ X such that f (x) = y. The function is bijective if the function is both one-to-one and onto.

Example 1.3. The function f (x) = 2x is one-to-one and onto from ℝ to ℝ. The fu...