1. Linear Equations (Review)
1.1 Introduction, Read on your own 1.2 Gaussian Elimination And Matrices, 1---5, 16 1.3 Gauss-Jordan Method, 1---3 1.4 Two-Point Boundary Value Problems, Read on your own 1.5 Making Gaussian Elimination Work, 1---3, 7 1.6 Ill-Conditioned Systems, 1, 2, 4 |
2. Echelon Forms (Review)
2.1 Row Echelon Form And Rank, 1---4 2.2 The Reduced Row Echelon Form, 1---3 2.3 Consistency Of Linear Systems, 1---7 2.4 Homogeneous Systems, 1---6 2.5 Nonhomogeneous Systems, 1---6 2.6 Electrical Circuits, Read on your own |
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3. Matrix Algebra (Review)
3.1 From ancient China to Arthur Cayley, Read on your own 3.2 Addition, Scalar Multiplication, And Transposition, Read on your own 3.3 Linearity, 1---2, 4 3.4 Why Do It This Way, Read on your own 3.5 Matrix Multiplication, Read on your own 3.6 Properties Of Matrix Multiplication, Read on your own 3.7 Matrix Inversion, 1---8, 10, 11 3.8 Inverses Of Sums and Sensitivity, 1---3 3.9 Elementary Matrices And Equivalence, 1---6 3.10 The LU Factorization, 1---3, 8, 9, 10 |
4. Vector Spaces
4.1 Spaces And Subspaces, 1---9 4.2 Four Fundamental Subspaces, 1---6, 9 11, 12, 13 4.3 Linear Independence, 1---7, 9, 10, 11 4.4 Basis And Dimension, 1---6, 8, 9, 16 4.5 More About Rank, 1---8, 9, 16, 20 4.6 Classical Least Squares, 1---4, 6, 9, 10 4.7 Linear Transformations, 1---9, 11---13, 16, 17 4.8 Change Of Basis And Similarity, 1---6, 12 4.9 Invariant Subspaces, 1---4, 8 |
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5. Norms, Inner Products, and Orthogonality
5.1 Vector Norms, 1---6 5.2 Matrix Norms, 1---5 5.3 Inner Product Spaces, 1---5 5.4 Orthogonal Vectors, 1---7, 9, 10, 12---14 5.5 Gram-Schmidt Procedure, 1---6 5.6 Unitary and Orthogonal Matrices, 1---5, 7, 8, 10, 13, 17 5.7 Orthogonal Reduction, 1---4 5.8 Discrete Fourier Transform, 1---6, 8, 19 5.9 Complementary Subspaces, 1---4, 6 5.10 Range-Nullspace Decomposition, 1---6, 10---12 5.11 Orthogonal Decomposition, 1---6, 8, 10, 12 5.12 Singular Value Decomposition, 1, 2, 4, 9, 13, 14, 16, 17 5.13 Orthogonal Projection, 1---7, 9---12 5.14 Why Least Squares? Read on your own 5.15 Angles Between Subspaces, Read on your own |
6. Determinants (Review)
6.1 Determinants, 1---14 6.2 Additional Properties Of Determinants, 1---12 |
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7. Eigenvalues And Eigenvectors
7.1 Elementary Properties Of Eigensystems, 1---17 7.2 Diagonalization by Similarity Transformations, 1---14 7.3 Functions Of Diagonalizable Matrices, 1---14, 16---17 7.4 Systems Of Differential Equations, 1---4 7.5 Normal Matrices, 1---10 7.6 Positive Definite Matrices, 1, 4---7 7.7 Nilpotent Matrices And Jordan Structure, 1, 2, 7, 8 7.8 Jordan Form, 1---4 7.9 Functions of Nondiagonalizable Matrices, 1, 3---11, 13, 14 7.10 Difference Equations, Limits, Summability, 1, 2, 4, 6 7.11 Minimum Polynomials and Krylov Methods, 1---6 |
8. Perron-Frobenius Theory of Nonnegative Matrices
8.1 Introduction, Read on your own 8.2 Positive Matrices, 1---5, 7 8.3 Nonnegative Matrices, 1--3, 5---7 8.4 Stochastic Matrices and Markov Chains, 1---3, 8, 9 |