# Elementary Linear Algebra

The Saylor Foundation, Jan 10, 2012 - Mathematics - 433 pages
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