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Preview Basics of Algebra, Topology, and Differential Calculus

Algebra, Topology, Differential Calculus, and Optimization Theory For Computer Science and Machine Learning Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] c Jean Gallier (cid:13) October 30, 2022 2 Contents Contents 3 1 Introduction 19 2 Groups, Rings, and Fields 21 2.1 Groups, Subgroups, Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 I Linear Algebra 47 3 Vector Spaces, Bases, Linear Maps 49 3.1 Motivations: Linear Combinations, Linear Independence, Rank . . . . . . . 49 3.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 (cid:80) 3.3 Indexed Families; the Sum Notation a . . . . . . . . . . . . . . . . . . 64 i I i 3.4 Linear Independence, Subspaces . . . .∈ . . . . . . . . . . . . . . . . . . . . 70 3.5 Bases of a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.6 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.7 Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.8 Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.9 Linear Forms and the Dual Space . . . . . . . . . . . . . . . . . . . . . . . . 98 3.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4 Matrices and Linear Maps 111 4.1 Representation of Linear Maps by Matrices . . . . . . . . . . . . . . . . . . 111 4.2 Composition of Linear Maps and Matrix Multiplication . . . . . . . . . . . 116 4.3 Change of Basis Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.4 The Effect of a Change of Bases on Matrices . . . . . . . . . . . . . . . . . 125 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5 Haar Bases, Haar Wavelets, Hadamard Matrices 137 3 4 CONTENTS 5.1 Introduction to Signal Compression Using Haar Wavelets . . . . . . . . . . 137 5.2 Haar Matrices, Scaling Properties of Haar Wavelets . . . . . . . . . . . . . . 139 5.3 Kronecker Product Construction of Haar Matrices . . . . . . . . . . . . . . 144 5.4 Multiresolution Signal Analysis with Haar Bases . . . . . . . . . . . . . . . 146 5.5 Haar Transform for Digital Images . . . . . . . . . . . . . . . . . . . . . . . 149 5.6 Hadamard Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6 Direct Sums 163 6.1 Sums, Direct Sums, Direct Products . . . . . . . . . . . . . . . . . . . . . . 163 6.2 Matrices of Linear Maps and Multiplication by Blocks . . . . . . . . . . . . 173 6.3 The Rank-Nullity Theorem; Grassmann’s Relation . . . . . . . . . . . . . . 186 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7 Determinants 201 7.1 Permutations, Signature of a Permutation . . . . . . . . . . . . . . . . . . . 201 7.2 Alternating Multilinear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.3 Definition of a Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 7.4 Inverse Matrices and Determinants . . . . . . . . . . . . . . . . . . . . . . . 218 7.5 Systems of Linear Equations and Determinants . . . . . . . . . . . . . . . . 221 7.6 Determinant of a Linear Map . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.7 The Cayley–Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.8 Permanents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.10 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 8 Gaussian Elimination, LU, Cholesky, Echelon Form 239 8.1 Motivating Example: Curve Interpolation . . . . . . . . . . . . . . . . . . . 239 8.2 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 8.3 Elementary Matrices and Row Operations . . . . . . . . . . . . . . . . . . . 248 8.4 LU-Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 8.5 PA = LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 (cid:126) 8.6 Proof of Theorem 8.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 8.7 Dealing with Roundoff Errors; Pivoting Strategies . . . . . . . . . . . . . . . 270 8.8 Gaussian Elimination of Tridiagonal Matrices . . . . . . . . . . . . . . . . . 272 8.9 SPD Matrices and the Cholesky Decomposition . . . . . . . . . . . . . . . . 274 8.10 Reduced Row Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 8.11 RREF, Free Variables, Homogeneous Systems . . . . . . . . . . . . . . . . . 289 8.12 Uniqueness of RREF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 8.13 Solving Linear Systems Using RREF . . . . . . . . . . . . . . . . . . . . . . 294 CONTENTS 5 8.14 Elementary Matrices and Columns Operations . . . . . . . . . . . . . . . . 300 (cid:126) 8.15 Transvections and Dilatations . . . . . . . . . . . . . . . . . . . . . . . . 301 8.16 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 8.17 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 9 Vector Norms and Matrix Norms 319 9.1 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 9.2 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 9.3 Subordinate Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 9.4 Inequalities Involving Subordinate Norms . . . . . . . . . . . . . . . . . . . 343 9.5 Condition Numbers of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 345 9.6 An Application of Norms: Inconsistent Linear Systems . . . . . . . . . . . . 354 9.7 Limits of Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . 355 9.8 The Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 9.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 9.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 10 Iterative Methods for Solving Linear Systems 369 10.1 Convergence of Sequences of Vectors and Matrices . . . . . . . . . . . . . . 369 10.2 Convergence of Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . 372 10.3 Methods of Jacobi, Gauss–Seidel, and Relaxation . . . . . . . . . . . . . . . 374 10.4 Convergence of the Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 10.5 Convergence Methods for Tridiagonal Matrices . . . . . . . . . . . . . . . . 385 10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 10.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 11 The Dual Space and Duality 395 11.1 The Dual Space E and Linear Forms . . . . . . . . . . . . . . . . . . . . . 395 ∗ 11.2 Pairing and Duality Between E and E . . . . . . . . . . . . . . . . . . . . 402 ∗ 11.3 The Duality Theorem and Some Consequences . . . . . . . . . . . . . . . . 407 11.4 The Bidual and Canonical Pairings . . . . . . . . . . . . . . . . . . . . . . . 413 11.5 Hyperplanes and Linear Forms . . . . . . . . . . . . . . . . . . . . . . . . . 415 11.6 Transpose of a Linear Map and of a Matrix . . . . . . . . . . . . . . . . . . 416 11.7 Properties of the Double Transpose . . . . . . . . . . . . . . . . . . . . . . . 423 11.8 The Four Fundamental Subspaces . . . . . . . . . . . . . . . . . . . . . . . 425 11.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 11.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 12 Euclidean Spaces 433 12.1 Inner Products, Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . 433 12.2 Orthogonality and Duality in Euclidean Spaces . . . . . . . . . . . . . . . . 442 12.3 Adjoint of a Linear Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 12.4 Existence and Construction of Orthonormal Bases . . . . . . . . . . . . . . 452 6 CONTENTS 12.5 Linear Isometries (Orthogonal Transformations) . . . . . . . . . . . . . . . . 459 12.6 The Orthogonal Group, Orthogonal Matrices . . . . . . . . . . . . . . . . . 462 12.7 The Rodrigues Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 12.8 QR-Decomposition for Invertible Matrices . . . . . . . . . . . . . . . . . . . 467 12.9 Some Applications of Euclidean Geometry . . . . . . . . . . . . . . . . . . . 472 12.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 12.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 13 QR-Decomposition for Arbitrary Matrices 487 13.1 Orthogonal Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 13.2 QR-Decomposition Using Householder Matrices . . . . . . . . . . . . . . . . 492 13.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 13.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 14 Hermitian Spaces 509 14.1 Hermitian Spaces, Pre-Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . 509 14.2 Orthogonality, Duality, Adjoint of a Linear Map . . . . . . . . . . . . . . . 518 14.3 Linear Isometries (Also Called Unitary Transformations) . . . . . . . . . . . 523 14.4 The Unitary Group, Unitary Matrices . . . . . . . . . . . . . . . . . . . . . 525 14.5 Hermitian Reflections and QR-Decomposition . . . . . . . . . . . . . . . . . 528 14.6 Orthogonal Projections and Involutions . . . . . . . . . . . . . . . . . . . . 533 14.7 Dual Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 14.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 14.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 15 Eigenvectors and Eigenvalues 549 15.1 Eigenvectors and Eigenvalues of a Linear Map . . . . . . . . . . . . . . . . . 549 15.2 Reduction to Upper Triangular Form . . . . . . . . . . . . . . . . . . . . . . 557 15.3 Location of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 15.4 Conditioning of Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . 565 15.5 Eigenvalues of the Matrix Exponential . . . . . . . . . . . . . . . . . . . . . 567 15.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 15.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 16 Unit Quaternions and Rotations in SO(3) 581 H 16.1 The Group SU(2) and the Skew Field of Quaternions . . . . . . . . . . . 581 16.2 Representation of Rotation in SO(3) By Quaternions in SU(2) . . . . . . . 583 16.3 Matrix Representation of the Rotation r . . . . . . . . . . . . . . . . . . . 588 q 16.4 An Algorithm to Find a Quaternion Representing a Rotation . . . . . . . . 590 16.5 The Exponential Map exp: su(2) SU(2) . . . . . . . . . . . . . . . . . . 593 → (cid:126) 16.6 Quaternion Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 16.7 Nonexistence of a “Nice” Section from SO(3) to SU(2) . . . . . . . . . . . . 597 16.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 CONTENTS 7 16.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 17 Spectral Theorems 603 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 17.2 Normal Linear Maps: Eigenvalues and Eigenvectors . . . . . . . . . . . . . . 603 17.3 Spectral Theorem for Normal Linear Maps . . . . . . . . . . . . . . . . . . . 609 17.4 Self-Adjoint and Other Special Linear Maps . . . . . . . . . . . . . . . . . . 614 17.5 Normal and Other Special Matrices . . . . . . . . . . . . . . . . . . . . . . . 620 17.6 Rayleigh–Ritz Theorems and Eigenvalue Interlacing . . . . . . . . . . . . . 623 17.7 The Courant–Fischer Theorem; Perturbation Results . . . . . . . . . . . . . 628 17.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 17.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 18 Computing Eigenvalues and Eigenvectors 639 18.1 The Basic QR Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 18.2 Hessenberg Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 18.3 Making the QR Method More Efficient Using Shifts . . . . . . . . . . . . . 653 18.4 Krylov Subspaces; Arnoldi Iteration . . . . . . . . . . . . . . . . . . . . . . 658 18.5 GMRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 18.6 The Hermitian Case; Lanczos Iteration . . . . . . . . . . . . . . . . . . . . . 663 18.7 Power Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 18.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 18.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 19 Introduction to The Finite Elements Method 669 19.1 A One-Dimensional Problem: Bending of a Beam . . . . . . . . . . . . . . . 669 19.2 A Two-Dimensional Problem: An Elastic Membrane . . . . . . . . . . . . . 680 19.3 Time-Dependent Boundary Problems . . . . . . . . . . . . . . . . . . . . . . 683 20 Graphs and Graph Laplacians; Basic Facts 691 20.1 Directed Graphs, Undirected Graphs, Weighted Graphs . . . . . . . . . . . 694 20.2 Laplacian Matrices of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 701 20.3 Normalized Laplacian Matrices of Graphs . . . . . . . . . . . . . . . . . . . 705 20.4 Graph Clustering Using Normalized Cuts . . . . . . . . . . . . . . . . . . . 709 20.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 20.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 21 Spectral Graph Drawing 715 21.1 Graph Drawing and Energy Minimization . . . . . . . . . . . . . . . . . . . 715 21.2 Examples of Graph Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . 718 21.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 22 Singular Value Decomposition and Polar Form 725 8 CONTENTS 22.1 Properties of f f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 ∗ ◦ 22.2 Singular Value Decomposition for Square Matrices . . . . . . . . . . . . . . 731 22.3 Polar Form for Square Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 735 22.4 Singular Value Decomposition for Rectangular Matrices . . . . . . . . . . . 737 22.5 Ky Fan Norms and Schatten Norms . . . . . . . . . . . . . . . . . . . . . . 741 22.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 22.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 23 Applications of SVD and Pseudo-Inverses 747 23.1 Least Squares Problems and the Pseudo-Inverse . . . . . . . . . . . . . . . . 747 23.2 Properties of the Pseudo-Inverse . . . . . . . . . . . . . . . . . . . . . . . . 754 23.3 Data Compression and SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . 759 23.4 Principal Components Analysis (PCA) . . . . . . . . . . . . . . . . . . . . . 761 23.5 Best Affine Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 772 23.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776 23.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 II Affine and Projective Geometry 781 24 Basics of Affine Geometry 783 24.1 Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 24.2 Examples of Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792 24.3 Chasles’s Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793 24.4 Affine Combinations, Barycenters . . . . . . . . . . . . . . . . . . . . . . . . 794 24.5 Affine Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799 24.6 Affine Independence and Affine Frames . . . . . . . . . . . . . . . . . . . . . 805 24.7 Affine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811 24.8 Affine Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818 24.9 Affine Geometry: A Glimpse . . . . . . . . . . . . . . . . . . . . . . . . . . 820 24.10 Affine Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824 24.11 Intersection of Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 826 25 Embedding an Affine Space in a Vector Space 829 25.1 The “Hat Construction,” or Homogenizing . . . . . . . . . . . . . . . . . . . 829 ˆ 25.2 Affine Frames of E and Bases of E . . . . . . . . . . . . . . . . . . . . . . . 836 ˆ 25.3 Another Construction of E . . . . . . . . . . . . . . . . . . . . . . . . . . . 839 25.4 Extending Affine Maps to Linear Maps . . . . . . . . . . . . . . . . . . . . . 842 26 Basics of Projective Geometry 847 26.1 Why Projective Spaces? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 26.2 Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852 26.3 Projective Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 CONTENTS 9 26.4 Projective Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 860 26.5 Projective Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874 26.6 Finding a Homography Between Two Projective Frames . . . . . . . . . . . 880 26.7 Affine Patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893 26.8 Projective Completion of an Affine Space . . . . . . . . . . . . . . . . . . . 896 26.9 Making Good Use of Hyperplanes at Infinity . . . . . . . . . . . . . . . . . 901 26.10 The Cross-Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904 26.11 Fixed Points of Homographies and Homologies . . . . . . . . . . . . . . . . 908 26.12 Duality in Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 922 26.13 Cross-Ratios of Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . 926 26.14 Complexification of a Real Projective Space . . . . . . . . . . . . . . . . . . 928 26.15 Similarity Structures on a Projective Space . . . . . . . . . . . . . . . . . . 930 26.16 Some Applications of Projective Geometry . . . . . . . . . . . . . . . . . . . 939 III The Geometry of Bilinear Forms 945 27 The Cartan–Dieudonn´e Theorem 947 27.1 The Cartan–Dieudonn´e Theorem for Linear Isometries . . . . . . . . . . . . 947 27.2 Affine Isometries (Rigid Motions) . . . . . . . . . . . . . . . . . . . . . . . . 959 27.3 Fixed Points of Affine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 961 27.4 Affine Isometries and Fixed Points . . . . . . . . . . . . . . . . . . . . . . . 963 27.5 The Cartan–Dieudonn´e Theorem for Affine Isometries . . . . . . . . . . . . 969 28 Isometries of Hermitian Spaces 973 28.1 The Cartan–Dieudonn´e Theorem, Hermitian Case . . . . . . . . . . . . . . . 973 28.2 Affine Isometries (Rigid Motions) . . . . . . . . . . . . . . . . . . . . . . . . 982 29 The Geometry of Bilinear Forms; Witt’s Theorem 987 29.1 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 29.2 Sesquilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995 29.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999 29.4 Adjoint of a Linear Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004 29.5 Isometries Associated with Sesquilinear Forms . . . . . . . . . . . . . . . . . 1006 29.6 Totally Isotropic Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 29.7 Witt Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016 29.8 Symplectic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 29.9 Orthogonal Groups and the Cartan–Dieudonn´e Theorem . . . . . . . . . . . 1028 29.10 Witt’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035 10 CONTENTS IV Algebra: PID’s, UFD’s, Noetherian Rings, Tensors, Modules over a PID, Normal Forms 1041 30 Polynomials, Ideals and PID’s 1043 30.1 Multisets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043 30.2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044 30.3 Euclidean Division of Polynomials . . . . . . . . . . . . . . . . . . . . . . . 1050 30.4 Ideals, PID’s, and Greatest Common Divisors . . . . . . . . . . . . . . . . . 1052 30.5 Factorization and Irreducible Factors in K[X] . . . . . . . . . . . . . . . . . 1060 30.6 Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064 30.7 Polynomial Interpolation (Lagrange, Newton, Hermite) . . . . . . . . . . . . 1071 31 Annihilating Polynomials; Primary Decomposition 1079 31.1 Annihilating Polynomials and the Minimal Polynomial . . . . . . . . . . . . 1081 31.2 Minimal Polynomials of Diagonalizable Linear Maps . . . . . . . . . . . . . 1083 31.3 Commuting Families of Linear Maps . . . . . . . . . . . . . . . . . . . . . . 1086 31.4 The Primary Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . 1089 31.5 Jordan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095 31.6 Nilpotent Linear Maps and Jordan Form . . . . . . . . . . . . . . . . . . . . 1098 31.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104 31.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105 32 UFD’s, Noetherian Rings, Hilbert’s Basis Theorem 1109 32.1 Unique Factorization Domains (Factorial Rings) . . . . . . . . . . . . . . . . 1109 32.2 The Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . 1123 32.3 Noetherian Rings and Hilbert’s Basis Theorem . . . . . . . . . . . . . . . . 1129 32.4 Futher Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133 33 Tensor Algebras 1135 33.1 Linear Algebra Preliminaries: Dual Spaces and Pairings . . . . . . . . . . . 1137 33.2 Tensors Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1142 33.3 Bases of Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154 33.4 Some Useful Isomorphisms for Tensor Products . . . . . . . . . . . . . . . . 1155 33.5 Duality for Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159 33.6 Tensor Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165 33.7 Symmetric Tensor Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172 33.8 Bases of Symmetric Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176 33.9 Some Useful Isomorphisms for Symmetric Powers . . . . . . . . . . . . . . . 1179 33.10 Duality for Symmetric Powers . . . . . . . . . . . . . . . . . . . . . . . . . . 1179 33.11 Symmetric Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183 33.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186 34 Exterior Tensor Powers and Exterior Algebras 1189

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Basics of Algebra, Topology, and Differential. Calculus. Jean Gallier. Department of Computer and Information Science. University of Pennsylvania.
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