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List of Symbols | |
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Vectors in R<sup>n</sup> and C<sup>n</sup>, Spatial Vectors | |
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Introduction | |
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Vectors in R<sup>n</sup> | |
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Vector Addition and Scalar Multiplication | |
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Dot (Inner) Product | |
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Located Vectors, Hyperplanes, Lines, Curves in R<sup>n</sup> | |
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Vectors in R<sup>3</sup> (Spatial Vectors), ijk Notation | |
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Complex Numbers | |
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Vectors in C<sup>n</sup> | |
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Algebra of Matrices | |
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Introduction | |
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Matrices | |
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Matrix Addition and Scalar Multiplication | |
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Summation Symbol | |
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Matrix Multiplication | |
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Transpose of a Matrix | |
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Square Matrices | |
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Powers of Matrices, Polynomials in Matrices | |
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Invertible (Nonsingular) Matrices | |
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Special Types of Square Matrices | |
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Complex Matrices | |
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Block Matrices | |
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Systems of Linear Equations | |
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Introduction | |
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Basic Definitions, Solutions | |
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Equivalent Systems, Elementary Operations | |
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Small Square Systems of Linear Equations | |
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Systems in Triangular and Echelon Forms | |
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Gaussian Elimination | |
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Echelon Matrices, Row Canonical Form, Row Equivalence | |
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Gaussian Elimination, Matrix Formulation | |
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Matrix Equation of a System of Linear Equations | |
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Systems of Linear Equations and Linear Combinations of Vectors | |
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Homogeneous Systems of Linear Equations | |
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Elementary Matrices | |
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LU Decomposition | |
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Vector Spaces | |
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Introduction | |
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Vector Spaces | |
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Examples of Vector Spaces | |
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Linear Combinations, Spanning Sets | |
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Subspaces | |
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Linear Spans, Row Space of a Matrix | |
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Linear Dependence and Independence | |
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Basis and Dimension | |
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Application to Matrices, Rank of a Matrix | |
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Sums and Direct Sums | |
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Coordinates | |
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Linear Mappings | |
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Introduction | |
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Mappings, Functions | |
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Linear Mappings (Linear Transformations) | |
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Kernel and Image of a Linear Mapping | |
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Singular and Nonsingular Linear Mappings, Isomorphisms | |
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Operations with Linear Mappings | |
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Algebra A(V) of Linear Operators | |
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Linear Mappings and Matrices | |
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Introduction | |
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Matrix Representation of a Linear Operator | |
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Change of Basis | |
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Similarity | |
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Matrices, and General Linear Mappings | |
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Inner Product Spaces, Orthogonality | |
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Introduction | |
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Inner Product Spaces | |
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Examples of Inner Product Spaces | |
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Cauchy-Schwarz Inequality, Applications | |
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Orthogonality | |
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Orthogonal Sets and Bases | |
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Gram-Schmidt Orthogonalization Process | |
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Orthogonal and Positive Definite Matrices | |
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Complex Inner Product Spaces | |
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Normed Vector Spaces (Optional) | |
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Determinants | |
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Introduction | |
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Determinants of Orders 1 and 2 | |
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Determinants of Order 3 | |
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Permutations | |
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Determinants of Arbitrary Order | |
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Properties of Determinants | |
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Minors and Cofactors | |
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Evaluation of Determinants | |
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Classical Adjoint | |
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Applications to Linear Equations, Cramer's Rule | |
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Submatrices, Minors, Principal Minors | |
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Block Matrices and Determinants | |
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Determinants and Volume | |
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Determinant of a Linear Operator | |
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Multilinearity and Determinants | |
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Diagonalization: Eigenvalues and Eigenvectors | |
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Introduction | |
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Polynomials of Matrices | |
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Characteristic Polynomial, Cayley-Hamilton Theorem | |
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Diagonalization, Eigenvalues and Eigenvectors | |
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Computing Eigenvalues and Eigenvectors, Diagonalizing Matrices | |
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Diagonalizing Real Symmetric Matrices and Quadratic Forms | |
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Minimal Polynomial | |
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Characteristic and Minimal Polynomials of Block Matrices | |
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Canonical Forms | |
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Introduction | |
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Triangular Form | |
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Invariance | |
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Invariant Direct-Sum Decompositions | |
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Primary Decomposition | |
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Nilpotent Operators | |
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Jordan Canonical Form | |
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Cyclic Subspaces | |
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Rational Canonical Form | |
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Quotient Spaces | |
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Linear Functionals and the Dual Space | |
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Introduction | |
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Linear Functionals and the Dual Space | |
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Dual Basis | |
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Second Dual Space | |
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Annihilators | |
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Transpose of a Linear Mapping | |
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Bilinear, Quadratic, and Hermitian Forms | |
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Introduction | |
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Bilinear Forms | |
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Bilinear Forms and Matrices | |
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Alternating Bilinear Forms | |
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Symmetric Bilinear Forms, Quadratic Forms | |
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Real Symmetric Bilinear Forms, Law of Inertia | |
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Hermitian Forms | |
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Linear Operators on Inner Product Spaces | |
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Introduction | |
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Adjoint Operators | |
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Analogy Between A(V) and C, Special Linear Operators | |
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Self-Adjoint Operators | |
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Orthogonal and Unitary Operators | |
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Orthogonal and Unitary Matrices | |
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Change of Orthonormal Basis | |
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Positive Definite and Positive Operators | |
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Diagonalization and Canonical Forms in Inner Product Spaces | |
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Spectral Theorem | |
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Multilinear Products | |
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Algebraic Structures | |
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Polynomials over a Field | |
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Odds and Ends | |
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Index | |