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A Guide to Advanced Linear Algebra

A Guide to Advanced Linear Algebra

Steven H. Weintraub
Volume: 44
Copyright Date: 2011
Edition: 1
Pages: 266
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  • Book Info
    A Guide to Advanced Linear Algebra
    Book Description:

    Linear algebra occupies a central place in modern mathematics. This book provides a rigorous and thorough development of linear algebra at an advanced level, and is directed at graduate students and professional mathematicians. It approaches linear algebra from an algebraic point of view, but its selection of topics is governed not only for their importance in linear algebra itself, but also for their applications throughout mathematics. Students in algebra, analysis, and topology will find much of interest and use to them, and the careful treatment and breadth of subject matter will make this book a valuable reference for mathematicians throughout their professional lives. Topics treated in this book include: vector spaces and linear transformations; dimension counting and applications; representation of linear transformations by matrices; duality; determinants and their uses; rational and especially Jordan canonical form; bilinear forms; inner product spaces; normal linear transformations and the spectral theorem; and an introduction to matrix groups as Lie groups. The book treats vector spaces in full generality, though it concentrates on the finite dimensional case. Also, it treats vector spaces over arbitrary fields, specializing to algebraically closed fields or to the fields of real and complex numbers as necessary.

    eISBN: 978-0-88385-967-4
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. I-VI)
  2. Preface
    (pp. VII-X)
    Steven H. Weintraub
  3. Table of Contents
    (pp. XI-XII)
  4. [Illustration]
    (pp. XIII-XIV)
  5. CHAPTER 1 Vector spaces and linear transformations
    (pp. 1-40)

    In this chapter we introduce the objects we will be studying and investigate some of their basic properties.

    Definition 1.1.1. A vector spaceVover a field 𝔽 is a setVwith a pair of operations (u,υ) ↦u+υforu,υϵVand (c,u) ↦cuforcϵ 𝔽,υϵVsatisfying the following axioms:

    (1)u+υϵVfor anyu,υϵV.

    (2)u+υ=υ+ufor anyu,υϵV.

    (3)u+ (υ+w) = (u...

  6. CHAPTER 2 Coordinates
    (pp. 41-56)

    In this chapter we investigate coordinates.

    It is useful to keep in mind the metaphor:

    Coordinates are a language for describing vectors and linear transformations.

    In human languages we have, for example:

    [*]English= star, [*]French= étoile, [*]German= Stern, [→]English= arrow, [→]French= flèche, [→]German= Pfeil.

    Coordinates share two similarities with human languages, but have one important difference.

    (1) Often it is easier to work with objects, and often it is easier to work with words that describe them. Similarly, often it is easier and more enlightening to work with vectors and linear transformations directly, and...

  7. CHAPTER 3 Determinants
    (pp. 57-88)

    In this chapter we deal with the determinant of a square matrix. The determinant has a simple geometric meaning, that of signed volume, and we use that to develop it in Section 3.1. We then present a more traditional and fuller development in Section 3.2. In Section 3.3 we derive important and useful properties of the determinant. In Section 3.4 we consider integrality questions, e.g., the question of the existence of integer (not just rational) solutions of the linear systemAx=b, a question best answered using determinants. In Section 3.5 we consider orientations, and see how to explain...

  8. CHAPTER 4 The structure of a linear transformation I
    (pp. 89-108)

    In this chapter we begin our analysis of the structure of a linear transformation 𝒯 :VV, whereVis a finite-dimensional 𝔽-vector space.

    We have arranged our exposition in order to bring some of the most important concepts to the fore first. Thus we begin with the notions of eigenvalues and eigenvectors, and we introduce the characteristic and minimum polynomials of a linear transformation early in this chapter as well. In this way we can get to some of the most important structural results, including results on diagonalizability and the Cayley-Hamilton theorem, as quickly as possible....

  9. CHAPTER 5 The structure of a linear transformation II
    (pp. 109-164)

    In this chapter we conclude our analysis of the structure of a linear transformation 𝒯 :VV. We derive our deepest structural results, the rational canonical form of 𝒯 and, whenVis a vector space over an algebraically closed field 𝔽, the Jordan canonical form of 𝒯 .

    Recall our metaphor of coordinates as giving a language in which to describe linear transformations. A basis 𝓑 ofVin which [𝒯]𝓑is in canonical form is a “right” language to describe the linear transformation 𝒯 . This is especially true for the Jordan canonical form, which...

  10. CHAPTER 6 Bilinear, sesquilinear, and quadratic forms
    (pp. 165-188)

    In this chapter we investigate bilinear, sesquilinear, and quadratic forms, or “forms” for short. A form is an additional structure on a vector space. Forms are interesting in their own right, and they have applications throughout mathematics. Many important vector spaces naturally come equipped with a form.

    In the first section we introduce forms and derive their basic properties. In the second section we see how to simplify forms on finite-dimensional vector spaces and in some cases completely classify them. In the third section we see how the presence of nonsingular form(s) enables us to define the adjoint of a...

  11. CHAPTER 7 Real and complex inner product spaces
    (pp. 189-222)

    In this chapter we consider real and complex vector spaces equipped with an inner product. An inner product is a special case of a symmetric bilinear form, in the real case, or of a Hermitian form, in the complex case. But it is a very important special case, one in which much more can be said than in general.

    We begin by defining the objects we will be studying.

    Definition 7.1.1. Aninner productφ(x,y) = ⟨x,y⟩ on a real vector spaceVis a symmetric bilinear form with the property that ⟨υ,υ⟩ > 0 for everyυ...

  12. CHAPTER 8 Matrix groups as Lie groups
    (pp. 223-230)

    Lie groups are central objects in mathematics. They lie at the intersection of algebra, analysis, and topology. In this chapter, we will show that many of the groups we have already encountered are in fact Lie groups.

    This chapter presupposes a certain knowledge of differential topology, and so we will use definitions and theorems from differential topology without further comment. We will also be a bit sketchy in our arguments in places. Throughout this chapter, “smooth” meansC. We usecijto denote a matrix entry that may be real or complex,xijto denote a real matrix entry and...

  13. CHAPTER A Polynomials
    (pp. 231-240)
  14. CHAPTER B Modules over principal ideal domains
    (pp. 241-244)
  15. Bibliography
    (pp. 245-246)
  16. Index
    (pp. 247-250)
  17. About the Author
    (pp. 251-251)