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Preface to Advanced Linear Algebra PDF Book
What is linear algebra, and how is it used? Upon examining almost any introductory text on linear algebra, we find a standard list of topics that seems to define the subject. (Advanced Linear Algebra by Nicholas Loehr)
On one hand, one part of linear algebra consists of computational techniques for solving linear equations, multiplying and inverting matrices, calculating and interpreting determinants, finding eigenvalues and eigenvectors, and so on.
On the other hand, there is a theoretical side to linear algebra involving abstract vector spaces, subspaces, linear independence, spanning sets, bases, dimension, and linear transformations.
But there is much more to linear algebra than just vector spaces, matrices, and linear equations! (Advanced Linear Algebra by Nicholas Loehr)
The goal of this Advanced Linear Algebra by Nicholas Loehr PDF book is to explore a variety of advanced topics in linear algebra, which highlight the rich interconnections linking this subject to geometry, algebra, analysis, combinatorics, numerical computation, and many other areas of mathematics.
The book consists of twenty chapters, grouped into six main subject areas (algebraic structures, matrices, structured matrices, geometric aspects of linear algebra, modules, and multilinear algebra).
Some chapters approach introductory material from a more sophisticated or abstract viewpoint; other chapters provide elementary expositions of more theoretical concepts, yet other chapters offer unusual perspectives or novel treatments of standard results.
Unlike some advanced mathematical texts, this Advanced Linear Algebra by Nicholas Loehr PDF Book has been carefully designed to minimize the dependence of each chapter on material found in earlier chapters.
Each chapter has been conceived as a “mathematical vignette” devoted to the development of one specific topic. (Advanced Linear Algebra by Nicholas Loehr)
If you need to learn about Jordan canonical forms, or ruler and compass constructions, or the singular value decomposition, or Hilbert spaces, or QR factorizations, or convexity, or normal matrices, or modules.
You may turn immediately to the relevant chapter without first wading through ten chapters of algebraic background or worrying that you will need a theorem covered two hundred pages earlier.
We do assume the reader has already encountered the basic linear algebra concepts described earlier (solving linear systems, computing with matrices and determinants, knowing elementary facts about vector spaces and linear maps).
These topics are all revisited in a more sophisticated setting at various points throughout the book, but this is not the Advanced Linear Algebra by Nicholas Loehr PDF Book you should read to learn the mechanics of Gaussian elimination or matrix multiplication for the first time!
Chapter 1 provides a condensed review of some pertinent definitions from abstract algebra. But the vast majority of the Advanced Linear Algebra by Nicholas Loehr PDF Book requires very little knowledge of abstract algebra beyond the definitions of fields.
Vector spaces over a field, subspaces, linear transformations, linear independence, and bases. (Advanced Linear Algebra by Nicholas Loehr)
The last three chapters build on the material on modules covered in Chapter 17, and in a few places, one needs to know about permutations (covered in Chapter 2) and polynomials (discussed in Chapter 3).
Although this Advanced Linear Algebra by Nicholas Loehr PDF Book focuses on theoretical aspects of linear algebra, giving complete proofs of all results.
We supplement and explain the general theory with many specific examples and concrete computations. (Advanced Linear Algebra by Nicholas Loehr)
The level of abstraction gradually increases as one proceeds through the book, as we move from matrices to vector spaces to modules.
Except in Chapter 16, we have deliberately avoided presenting the mathematical content as a dry skeleton of itemized definitions, theorems, and proofs. Instead.
The material is presented in a narrative format, providing motivation, examples, and informal discussions to help the reader understand the significance of the theorems and the intuition behind the proofs.
Each chapter ends with a summary containing a structured list of the principal definitions and results covered in the chapter, followed by a set of exercises.
Most exercises are not used in the main text and consist of examples, computations, and proofs that will aid the reader in assimilating the material from the chapter.
The reader is encouraged to use a computer algebra system to help solve computationally intensive exercises.
This text is designed for use by advanced undergraduates or beginning graduate students, or as a reference. (Advanced Linear Algebra by Nicholas Loehr)
For the second course in linear algebra, one could cover most of the chapters in Parts II, III, and IV, reviewing Part I as needed.
For the second course in abstract algebra focusing on the role of linear-algebraic ideas, one could cover Parts I, V, and VI along with selections from the rest of the book (such as Chapter 12 or Chapter 13).
The next section describes the contents of each chapter; for more details on what is covered in a given chapter, consult the summary for that chapter. (Advanced Linear Algebra by Nicholas Loehr)
The Author of Advanced Linear Algebra
Nicholas Loehr is the editor of Advanced Linear Algebra PDF Book.
Main Contents of Advanced Linear Algebra PDF Book
- Overview of Algebraic Systems
- Basic Matrix Operations
- Determinants via Calculations
- Concrete vs. Abstract Linear Algebra
- Hermitian, Positive Definite, Unitary, and Normal Matrices
- Jordan Canonical Forms
- Matrix Factorizations
- Iterative Algorithms in Numerical Linear Algebra
- Affine Geometry and Convexity
- Ruler and Compass Constructions
- Dual Spaces and Bilinear Forms
- Metric Spaces and Hilbert Spaces
- Finitely Generated Commutative Groups
- Axiomatic Approach to Independence, Bases, and Dimension
- Elements of Module Theory
- Principal Ideal Domains, Modules over PIDs, and Canonical Forms
- Introduction to Universal Mapping Properties
- Universal Mapping Problems in Multilinear Algebra
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