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Applied linear algebra
Olver, Peter J

Cataloging brief

Applied linear algebra
Author :   Olver, Peter J
Publisher :   Springer International Publishing AG
Pub. Year  :   2018
Subjects :   Algebras, Linear Mathematics
Call Number :   ‭QA 184 .2 .O48 2018

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  • Preface (7)
    • Syllabi and Prerequisites (9)
    • Survey of Topics (10)
    • Course Outlines (12)
    • Comments on Individual Chapters (14)
    • Changes from the First Edition (15)
    • Exercises and Software (16)
    • Conventions and Notations (17)
    • History and Biography (19)
    • Some Final Remarks (19)
    • Acknowledgments (20)
  • Table of Contents (21)
  • Chapter 1: Linear Algebraic Systems (26)
    • 1.1 Solution of Linear Systems (26)
    • 1.2 Matrices and Vectors (28)
      • Matrix Arithmetic (30)
    • 1.3 Gaussian Elimination—Regular Case (37)
      • Elementary Matrices (41)
      • The LU Factorization (43)
      • Forward and Back Substitution (45)
    • 1.4 Pivoting and Permutations (47)
      • Permutations and Permutation Matrices (50)
      • The Permuted LU Factorization (52)
    • 1.5 Matrix Inverses (56)
      • Gauss–Jordan Elimination (60)
      • Solving Linear Systems with the Inverse (65)
      • The LDV Factorization (66)
    • 1.6 Transposes and Symmetric Matrices (68)
      • Factorization of Symmetric Matrices (70)
    • 1.7 Practical Linear Algebra (73)
      • Tridiagonal Matrices (77)
      • Pivoting Strategies (80)
    • 1.8 General Linear Systems (84)
      • Homogeneous Systems (92)
    • 1.9 Determinants (94)
  • Chapter 2: Vector Spaces and Bases (100)
    • 2.1 Real Vector Spaces (101)
    • 2.2 Subspaces (106)
    • 2.3 Span and Linear Independence (112)
      • Linear Independence and Dependence (117)
    • 2.4 Basis and Dimension (123)
    • 2.5 The Fundamental Matrix Subspaces (130)
      • Kernel and Image (130)
      • The Superposition Principle (135)
      • Adjoint Systems, Cokernel, and Coimage (137)
      • The Fundamental Theorem of Linear Algebra (139)
    • 2.6 Graphs and Digraphs (145)
  • Chapter 3: Inner Products and Norms (154)
    • 3.1 Inner Products (154)
      • Inner Products on Function Spaces (158)
    • 3.2 Inequalities (162)
      • The Cauchy–Schwarz Inequality (162)
      • Orthogonal Vectors (165)
      • The Triangle Inequality (167)
    • 3.3 Norms (169)
      • Unit Vectors (173)
      • Equivalence of Norms (175)
      • Matrix Norms (178)
    • 3.4 Positive Definite Matrices (181)
      • Gram Matrices (186)
    • 3.5 Completing the Square (191)
      • The Cholesky Factorization (196)
    • 3.6 Complex Vector Spaces (197)
      • Complex Numbers (198)
      • Complex Vector Spaces and Inner Products (202)
  • Chapter 4: Orthogonality (208)
    • 4.1 Orthogonal and Orthonormal Bases (209)
      • Computations in Orthogonal Bases (213)
    • 4.2 The Gram–Schmidt Process (217)
      • Modifications of the Gram–Schmidt Process (222)
    • 4.3 Orthogonal Matrices (225)
      • The QR Factorization (230)
      • Ill-Conditioned Systems and Householder’s Method (233)
    • 4.4 Orthogonal Projections and Orthogonal Subspaces (237)
      • Orthogonal Projection (238)
      • Orthogonal Subspaces (241)
      • Orthogonality of the Fundamental Matrix Subspaces and the Fredholm Alternative (246)
    • 4.5 Orthogonal Polynomials (251)
      • The Legendre Polynomials (252)
      • Other Systems of Orthogonal Polynomials (256)
  • Chapter 5: Minimization and Least Squares (260)
    • 5.1 Minimization Problems (260)
      • Equilibrium Mechanics (261)
      • Solution of Equations (261)
      • The Closest Point (263)
    • 5.2 Minimization of Quadratic Functions (264)
    • 5.3 The Closest Point (270)
    • 5.4 Least Squares (275)
    • 5.5 Data Fitting and Interpolation (279)
      • Polynomial Approximation and Interpolation (284)
      • Approximation and Interpolation by General Functions (296)
      • Least Squares Approximation in Function Spaces (299)
      • Orthogonal Polynomials and Least Squares (302)
      • Splines (304)
    • 5.6 Discrete Fourier Analysis and the Fast Fourier Transform (310)
      • Compression and Denoising (318)
      • The Fast Fourier Transform (320)
  • Chapter 6: Equilibrium (326)
    • 6.1 Springs and Masses (326)
      • Positive Definiteness and the Minimization Principle (334)
    • 6.2 Electrical Networks (336)
      • Batteries, Power, and the Electrical–Mechanical Correspondence (342)
    • 6.3 Structures (347)
  • Chapter 7: Linearity (366)
    • 7.1 Linear Functions (367)
      • Linear Operators (372)
      • The Space of Linear Functions (374)
      • Dual Spaces (375)
      • Composition (377)
      • Inverses (380)
    • 7.2 Linear Transformations (383)
      • Change of Basis (390)
    • 7.3 Affine Transformations and Isometries (395)
      • Isometry (397)
    • 7.4 Linear Systems (401)
      • The Superposition Principle (403)
      • Inhomogeneous Systems (408)
      • Superposition Principles for Inhomogeneous Systems (413)
      • Complex Solutions to Real Systems (415)
    • 7.5 Adjoints, Positive Definite Operators, and Minimization Principles (420)
      • Self-Adjoint and Positive Definite Linear Functions (423)
      • Minimization (425)
  • Chapter 8: Eigenvalues and Singular Values (428)
    • 8.1 Linear Dynamical Systems (429)
      • Scalar Ordinary Differential Equations (429)
      • First Order Dynamical Systems (432)
    • 8.2 Eigenvalues and Eigenvectors (433)
      • Basic Properties of Eigenvalues (440)
      • The Gershgorin Circle Theorem (445)
    • 8.3 Eigenvector Bases (448)
      • Diagonalization (451)
    • 8.4 Invariant Subspaces (454)
    • 8.5 Eigenvalues of Symmetric Matrices (456)
      • The Spectral Theorem (462)
      • Optimization Principles for Eigenvalues of Symmetric Matrices (465)
    • 8.6 Incomplete Matrices (469)
      • The Schur Decomposition (469)
      • The Jordan Canonical Form (472)
    • 8.7 Singular Values (479)
      • The Pseudoinverse (482)
      • The Euclidean Matrix Norm (484)
      • Condition Number and Rank (485)
      • Spectral Graph Theory (487)
    • 8.8 Principal Component Analysis (492)
      • Variance and Covariance (492)
      • The Principal Components (496)
  • Chapter 9: Iteration (500)
    • 9.1 Linear Iterative Systems (501)
      • Scalar Systems (501)
      • Powers of Matrices (504)
      • Diagonalization and Iteration (509)
    • 9.2 Stability (513)
      • Spectral Radius (514)
      • Fixed Points (518)
      • Matrix Norms and Convergence (520)
    • 9.3 Markov Processes (524)
    • 9.4 Iterative Solution of Linear Algebraic Systems (531)
      • The Jacobi Method (533)
      • The Gauss–Seidel Method (537)
      • Successive Over-Relaxation (542)
    • 9.5 Numerical Computation of Eigenvalues (547)
      • The Power Method (547)
      • The QR Algorithm (551)
      • Tridiagonalization (557)
    • 9.6 Krylov Subspace Methods (561)
      • Krylov Subspaces (561)
      • Arnoldi Iteration (562)
      • The Full Orthogonalization Method (565)
      • The Conjugate Gradient Method (567)
      • The Generalized Minimal Residual Method (571)
    • 9.7 Wavelets (574)
      • The Haar Wavelets (574)
      • Modern Wavelets (580)
      • Solving the Dilation Equation (584)
  • Chapter 10: Dynamics (589)
    • 10.1 Basic Solution Techniques (589)
      • The Phase Plane (591)
      • Existence and Uniqueness (594)
      • Complete Systems (596)
      • The General Case (599)
    • 10.2 Stability of Linear Systems (603)
    • 10.3 Two-Dimensional Systems (609)
      • Distinct Real Eigenvalues (610)
      • Complex Conjugate Eigenvalues (611)
      • Incomplete Double Real Eigenvalue (612)
      • Complete Double Real Eigenvalue (612)
    • 10.4 Matrix Exponentials (616)
      • Applications in Geometry (623)
      • Invariant Subspaces and Linear Dynamical Systems (627)
      • Inhomogeneous Linear Systems (629)
    • 10.5 Dynamics of Structures (632)
      • Stable Structures (634)
      • Unstable Structures (639)
      • Systems with Differing Masses (642)
      • Friction and Damping (644)
    • 10.6 Forcing and Resonance (647)
      • Electrical Circuits (652)
      • Forcing and Resonance in Systems (654)
  • References (657)
  • Symbol Index (661)
  • Subject Index (666)
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