## Elementary Linear AlgebraThis 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

11 | |

23 | |

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 |

### Common terms and phrases

algebraic algorithm angle augmented matrix basis of eigenvectors called coefﬁcients complex numbers consider Corollary deﬁned deﬁnition denote determinant diagonal matrix dot product eigenvalues eigenvalues and eigenvectors eigenvectors equal to zero equals zero equivalent Example exists Explain ﬁlm ﬁnd Find the eigenvalues Find the matrix formula function geometric given Hint inner product space integer inverse ith row Lemma linear combination linear transformation linearly independent LU factorization main diagonal minimal polynomial nonnegative nonzero norm obtain original matrix orthogonal matrix orthonormal basis orthonormal set pivot column positive problem Proof rank real numbers reduced echelon form result rotates every vector row and add row operations row reduced echelon second row set of vectors Show simplex tableau subspace Suppose symmetric matrix system of equations Theorem transformation which rotates unique vector in R2 vector space verify yields