## Linear Algebra Done RightThis text for a second course in linear algebra is aimed at math majors and graduate students. The novel approach taken here banishes determinants to the end of the book and focuses on the central goal of linear algebra: understanding the structure of linear operators on vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. For example, the book presents--without having defined determinants--a clean proof that every linear operator on a finite-dimensional complex vector space (or an odd-dimensional real vector space) has an eigenvalue. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus, the text starts by discussing vector spaces, linear independence, span, basis, and dimension. Students are introduced to inner-product spaces in the first half of the book and shortly thereafter to the finite-dimensional spectral theorem. This second edition includes a new section on orthogonal projections and minimization problems. The sections on self-adjoint operators, normal operators, and the spectral theorem have been rewritten. New examples and new exercises have been added, several proofs have been simplified, and hundreds of minor improvements have been made throughout the text. |

### What people are saying - Write a review

#### For an analytic approach to Linear Algebra

User Review - oken - Overstock.comGood for a second course in Linear Algebra from an analysis oriented approach. Read full review

### Contents

Vector Spaces | 1 |

Complex Numbers | 2 |

Definition of Vector Space | 4 |

Properties of Vector Spaces | 11 |

Subspaces | 13 |

Sums and Direct Sums | 14 |

Exercises | 19 |

FiniteDimensional Vector Spaces | 21 |

Orthogonal Projections and Minimization Problems | 111 |

Linear Functionals and Adjoints | 117 |

Exercises | 122 |

Operators on InnerProduct Spaces | 127 |

SelfAdjoint and Normal Operators | 128 |

The Spectral Theorem | 132 |

Normal Operators on Real InnerProduct Spaces | 138 |

Positive Operators | 144 |

Span and Linear Independence | 22 |

Bases | 27 |

Dimension | 31 |

Exercises | 35 |

Linear Maps | 37 |

Definitions and Examples | 38 |

Null Spaces and Ranges | 41 |

The Matrix of a Linear Map | 48 |

Invertibility | 53 |

Exercises | 59 |

Polynomials | 63 |

Degree | 64 |

Complex Coefficients | 67 |

Real Coefficients | 69 |

Exercises | 73 |

Eigenvalues and Eigenvectors | 75 |

Invariant Subspaces | 76 |

Polynomials Applied to Operators | 80 |

UpperTriangular Matrices | 81 |

Diagonal Matrices | 87 |

Invariant Subspaces on Real Vector Spaces | 91 |

Exercises | 94 |

InnerProduct Spaces | 97 |

Inner Products | 98 |

Norms | 102 |

Orthonormal Bases | 106 |

Isometries | 147 |

Polar and SingularValue Decompositions | 152 |

Exercises | 158 |

Operators on Complex Sector Spaces | 163 |

Generalized Eigenvectors | 164 |

The Characteristic Polynomial | 168 |

Decomposition of an Operator | 173 |

Square Roots | 177 |

The Minimal Polynomial | 179 |

Jordan Form | 183 |

Exercises | 188 |

Operators on Real Vector Spaces | 193 |

Eigenvalues of Square Matrices | 194 |

Block UpperTriangular Matrices | 195 |

The Characteristic Polynomial | 198 |

Exercises | 210 |

Trace and Determinant | 213 |

Change of Basis | 214 |

Trace | 216 |

Determinant of an Operator | 222 |

Determinant of a Matrix | 225 |

Volume | 236 |

Exercises | 244 |

Symbol Index | 247 |

249 | |

### Other editions - View all

### Common terms and phrases

addition applied bases block called chapter characteristic polynomial choose Clearly coefficients column completing the proof complex vector space compute consider consisting contains coordinates corollary corresponding decomposition defined definition denote dependent desired determinant diagonal diagonal matrix dimension dimensional dimV distinct eigenpairs eigenvalues eigenvectors element entries equals equation example Exercise exists factorization finite formula function given gives hence holds identity implies injective inner invariant invertible isometry length linear algebra linear map linearly independent Mathematics matrix with respect means multiplication nilpotent nonnegative nonzero normal Note null Obviously operator orthogonal orthonormal basis permutation proof properties proposition Prove range real vector space Recall respect result scalar multiplication self-adjoint shows span square root standard basis subspace Suppose surjective term theorem trace unique upper-triangular matrix values verify volume write