Linear Algebra, Second Edition

Preface iii......Page f003.djvu
Contents vi......Page f006.djvu
1.1 Fields 1......Page p001.djvu
1.2 Systems of Linear Equations 3......Page p003.djvu
1.3 Matrices and Elementary Row Operations 6......Page p006.djvu
1.4 Row-Reduced Echelon Matrices 11......Page p011.djvu
1.5 Matrix Multiplication 16......Page p016.djvu
1.6 Invertible Matrices 21......Page p021.djvu
2.1 Vector Spaces 28......Page p028.djvu
2.2 Subspaces 34......Page p034.djvu
2.3 Bases and Dimension 40......Page p040.djvu
2.4 Coordinates 49......Page p049.djvu
2.5 Summary of Row-Equivalence 55......Page p055.djvu
2.6 Computations Concerning Subspaces 58......Page p058.djvu
3.1 Linear Transformations 67......Page p067.djvu
3.2 The Algebra of Linear Transformations 74......Page p074.djvu
3.3 Isomorphism 84......Page p084.djvu
3.4 Representation of Transformations by Matrices 86......Page p086.djvu
3.5 Linear Functionals 97......Page p097.djvu
3.6 The Double Dual 107......Page p107.djvu
3.7 The Transpose of a Linear Transformation 111......Page p111.djvu
4.1 Algebras 117......Page p117.djvu
4.2 The Algebra of Polynomials 119......Page p119.djvu
4.3 Lagrange Interpolation 124......Page p124.djvu
4.4 Polynomial Ideals 127......Page p127.djvu
4.5 The Prime Factorization of a Polynomial 134......Page p134.djvu
5.1 Commutative Rings 140......Page p140.djvu
5.2 Determinant Functions 141......Page p141.djvu
5.3 Permutations and the Uniqueness of Determinants 150......Page p150.djvu
5.4 Additional Properties of Determinants 156......Page p156.djvu
5.5 Modules 164......Page p164.djvu
5.6 Multilinear Functions 166......Page p166.djvu
5.7 The Grassman Ring 173......Page p173.djvu
6.1 Introduction 181......Page p181.djvu
6.2 Characteristic Values 182......Page p182.djvu
6.3 Annihilating Polynomials 190......Page p190.djvu
6.4 Invariant Subspaces 198......Page p198.djvu
6.5 Simultaneous Triangulation; Simultaneous Diagonalization 206......Page p206.djvu
6.6 Direct-Sum Decompositions 209......Page p209.djvu
6.7 Invariant Direct Sums 213......Page p213.djvu
6.8 The Primary Decomposition Theorem 219......Page p219.djvu
7.1 Cyclic Subspaces and Annihilators 227......Page p227.djvu
7.2 Cyclic Decompositions and the Rational Form 231......Page p231.djvu
7.3 The Jordan Form 244......Page p244.djvu
7.4 Computation of Invariant Factors 251......Page p251.djvu
7.5 Summary; Semi-Simple Operators 262......Page p262.djvu
8.1 Inner Products 270......Page p270.djvu
8.2 Inner Product Spaces 277......Page p277.djvu
8.3 Linear Functional and Adjoints 290......Page p290.djvu
8.4 Unitary Operators 299......Page p299.djvu
8.5 Normal Operators 311......Page p311.djvu
9.1 Introduction 319......Page p319.djvu
9.2 Forms on Inner Product Spaces 320......Page p320.djvu
9.3 Positive Forms 325......Page p325.djvu
9.4 More on Forms 332......Page p332.djvu
9.5 Spectral Theory 335......Page p335.djvu
9.6 Further Properties of Normal Operators 349......Page p349.djvu
10.1 Bilinear Forms 359......Page p359.djvu
10.2 Symmetric Bilinear Forms 367......Page p367.djvu
10.3 Skew-Symmetric Bilinear Forms 375......Page p375.djvu
10.4 Groups Preserving Bilinear Forms 379......Page p379.djvu
Appendix 386......Page p386.djvu
A.1 Sets 387......Page p387.djvu
A.2 Functions 388......Page p388.djvu
A.3 Equivalence Relations 391......Page p391.djvu
A.4 Quotient Spaces 394......Page p394.djvu
A.5 Equivalence Relations in Linear Algebra 397......Page p397.djvu
A.6 The Axiom of Choice 399......Page p399.djvu
Bibliography 400......Page p400.djvu
Index 401......Page p401.djvu