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1992 | Buch

Preliminaries Kapitel lesen Erstes Kapitel lesen

Advanced Linear Algebra

verfasst von: Steven Roman

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

This book is a thorough introduction to linear algebra, for the graduate or advanced undergraduate student. Prerequisites are limited to a knowledge of the basic properties of matrices and determinants. However, since we cover the basics of vector spaces and linear transformations rather rapidly, a prior course in linear algebra (even at the sophomore level), along with a certain measure of "mathematical maturity," is highly desirable. Chapter 0 contains a summary of certain topics in modern algebra that are required for the sequel. This chapter should be skimmed quickly and then used primarily as a reference. Chapters 1-3 contain a discussion of the basic properties of vector spaces and linear transformations. Chapter 4 is devoted to a discussion of modules, emphasizing a comparison between the properties of modules and those of vector spaces. Chapter 5 provides more on modules. The main goals of this chapter are to prove that any two bases of a free module have the same cardinality and to introduce noetherian modules. However, the instructor may simply skim over this chapter, omitting all proofs. Chapter 6 is devoted to the theory of modules over a principal ideal domain, establishing the cyclic decomposition theorem for finitely generated modules. This theorem is the key to the structure theorems for finite dimensional linear operators, discussed in Chapters 7 and 8. Chapter 9 is devoted to real and complex inner product spaces.

Inhaltsverzeichnis

Frontmatter

Preliminaries

Chapter 0. Preliminaries
Abstract
In this chapter, we briefly discuss some topics that are needed for the sequel. This chapter should be skimmed quickly and then used primarily as a reference.
Steven Roman

Basic Linear Algebra

Frontmatter
Chapter 1. Vector Spaces
Abstract
Let us begin with the definition of our principle object of study.
Steven Roman
Chapter 2. Linear Transformations
Abstract
Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more precise.
Steven Roman
Chapter 3. The Isomorphism Theorems
Abstract
Let S be a subspace of a vector space V, and let ≡ s be the binary relation on V defined by
$$u{ \equiv _s}v\; \Leftrightarrow u - v \in S$$
Steven Roman
Chapter 4. Modules I
Abstract
Let V be a vector space over a field F, and let τ ∈ L(V). Then for any polynomial p(x) ∈ F[x], the operator p(τ) is well-defined. For instance, if p(x) = 1 + 2x + x3, then
$$p\left( \tau \right) = l + 2\tau + {\tau ^3}$$
where ι is the identity operator, and τ 3 is the threefold composition τ o τ o τ.
Steven Roman
Chapter 5. Modules II
Abstract
The procedure for defining quotient modules is the same as that for defining quotient spaces. We summarize in the following theorem.
Steven Roman
Chapter 6. Modules over Principal Ideal Domains
Abstract
When a ring R has nice properties (such as being noetherian), then its R-modules tend to have nice properties (such as being noetherian, at least in the finitely generated case). Since principal ideal domains (abbreviated p.i.d.s) have very nice properties, we expect the same for modules over p.i.d.s.
Steven Roman
Chapter 7. The Structure of a Linear Operator
Abstract
In this chapter, we study the structure of a linear operator on a finite dimensional vector space, using the powerful module decomposition theorems of the previous chapter. Unless otherwise noted, all vector spaces will be assumed to be finite dimensional.
Steven Roman
Chapter 8. Eigenvalues and Eigenvectors
Abstract
Unless otherwise noted, we will assume throughout this chapter that all vector spaces are finite dimensional.
Steven Roman
Chapter 9. Real and Complex Inner Product Spaces
Abstract
We now turn to a discussion of real or complex vector apaces that have an additional function defined on them, called an inner product, as described in the upcoming definition. Thus in this chapter, F will denote either the real or complex field.
Steven Roman
Chapter 10. The Spectral Theorem for Normal Operators
Abstract
The purpose of this chapter is to study the structure of certain special types of linear operators on an inner product space. In order to define these operators, we introduce another type of adjoint (different from the operator adjoint of Chapter 3). We will define this adjoint in the finite dimensional case only, deferring the infinite dimensional case to Chapter 13.
Steven Roman

Topics

Frontmatter
Chapter 11. Metric Vector Spaces
Abstract
In this chapter, we study vector spaces over arbitrary fields that have a bilinear form defined on them. As we will see, the study of such vector spaces has a very geometric flavor, and hence so does the terminology.
Steven Roman
Chapter 12. Metric Spaces
Abstract
In Chapter 9, we studied the basic properties of real and complex inner product spaces. Much of what we did does not depend on whether the space in question is finite or infinite dimensional. However, as we discussed in Chapter 9, the presence of an inner product, and hence a metric, on a vector space, raises a host of new issues related to convergence. In this chapter, we discuss briefly the concept of a metric space. This will enable us to study the convergence properties of real and complex inner product spaces.
Steven Roman
Chapter 13. Hilbert Spaces
Abstract
Now that we have the necessary backgroung on the topological properties of metric spaces, we can resume our study of inner product spaces without qualification as to dimension. As in Chapter 9, we restrict attention to real and complex inner product spaces. Hence F will denote either ℝ or ℂ.
Steven Roman
Chapter 14. Tensor Products
Abstract
In the preceding chapters, we have seen several ways to construct new vector spaces from old ones. Two of the most important such constructions are the direct sum U ⊕ V and the set L(U,V) of all linear transformations from U to V. In this chapter, we consider another construction, known as the tensor product.
Steven Roman
Chapter 15. Affine Geometry
Abstract
In this chapter, we will study the geometry of a finite dimensional vector space V, along with its structure preserving maps. Throughout this chapter, all vector spaces are assumed to be finite dimensional.
Steven Roman
Chapter 16. The Umbral Calculus
Abstract
In this chapter, we give a brief introduction to a relatively new subject, called the umbral calculus. This is an algebraic theory used to study certain types of polynomial functions that play an important role in applied mathematics. We give only a brief introduction to the subject — emphasizing the algebraic aspects rather than the applications. For more on the umbral calculus, we suggest The Umbral Calculus, by Roman [1984].
Steven Roman
Backmatter
Metadaten
Titel
Advanced Linear Algebra
verfasst von
Steven Roman
Copyright-Jahr
1992
Verlag
Springer New York
Electronic ISBN
978-1-4757-2178-2
Print ISBN
978-1-4757-2180-5
DOI
https://doi.org/10.1007/978-1-4757-2178-2