Elementary Linear Algebra

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The Saylor Foundation, Jan 10, 2012 - Mathematics - 433 pages
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This is an introduction to linear algebra. The main part of the book features row operations and everything is done in terms of the row reduced echelon form and specific algorithms. At the end, the more abstract notions of vector spaces and linear transformations on vector spaces are presented. However, this is intended to be a first course in linear algebra for students who are sophomores or juniors who have had a course in one variable calculus and a reasonable background in college algebra. I have given complete proofs of all the fundamental ideas, but some topics such as Markov matrices are not complete in this book but receive a plausible introduction. The book contains a complete treatment of determinants and a simple proof of the Cayley Hamilton theorem although these are optional topics. The Jordan form is presented as an appendix. I see this theorem as the beginning of more advanced topics in linear algebra and not really part of a beginning linear algebra course. There are extensions of many of the topics of this book in my on line book. I have also not emphasized that linear algebra can be carried out with any field although there is an optional section on this topic, most of the book being devoted to either the real numbers or the complex numbers. It seems to me this is a reasonable specialization for a first course in linear algebra.
 

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Contents

Some Prerequisite Topics
11
Fn
23
Vector Products
39
Systems Of Equations
59
Matrices
77
Determinants
99
The Mathematical Theory Of Determinants
119
Rank Of A Matrix
131
Spectral Theory
217
Matrices And The Inner Product
247
Numerical Methods For Solving Linear Systems
285
Numerical Methods For Solving The Eigenvalue Problem
299
Vector Spaces
325
Linear Transformations
369
The Jordan Canonical Form
391
The Fundamental Theorem Of Algebra
399

Linear Transformations
165
The LU Factorization
181
Linear Programming
195
Answers To Selected Exercises
403
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