# Linear Algebra Done Right

Springer Science & Business Media, 18 Tem 1997 - 251 sayfa
This 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.

### İçindekiler

 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 Index 249 Telif Hakkı

### Bu kitaba yapılan referanslar

 Linear Algebra and Its ApplicationsPeter D. LaxSınırlı önizleme - 2007
 Lie Groups, Lie Algebras, and Representations: An Elementary IntroductionBrian C. HallSınırlı önizleme - 2003
Tüm Kitap Arama sonuçları &raquo;