| |

| |

List of Symbols | |

| |

| |

| |

Vectors in R<sup>n</sup> and C<sup>n</sup>, Spatial Vectors | |

| |

| |

| |

Introduction | |

| |

| |

| |

Vectors in R<sup>n</sup> | |

| |

| |

| |

Vector Addition and Scalar Multiplication | |

| |

| |

| |

Dot (Inner) Product | |

| |

| |

| |

Located Vectors, Hyperplanes, Lines, Curves in R<sup>n</sup> | |

| |

| |

| |

Vectors in R<sup>3</sup> (Spatial Vectors), ijk Notation | |

| |

| |

| |

Complex Numbers | |

| |

| |

| |

Vectors in C<sup>n</sup> | |

| |

| |

| |

Algebra of Matrices | |

| |

| |

| |

Introduction | |

| |

| |

| |

Matrices | |

| |

| |

| |

Matrix Addition and Scalar Multiplication | |

| |

| |

| |

Summation Symbol | |

| |

| |

| |

Matrix Multiplication | |

| |

| |

| |

Transpose of a Matrix | |

| |

| |

| |

Square Matrices | |

| |

| |

| |

Powers of Matrices, Polynomials in Matrices | |

| |

| |

| |

Invertible (Nonsingular) Matrices | |

| |

| |

| |

Special Types of Square Matrices | |

| |

| |

| |

Complex Matrices | |

| |

| |

| |

Block Matrices | |

| |

| |

| |

Systems of Linear Equations | |

| |

| |

| |

Introduction | |

| |

| |

| |

Basic Definitions, Solutions | |

| |

| |

| |

Equivalent Systems, Elementary Operations | |

| |

| |

| |

Small Square Systems of Linear Equations | |

| |

| |

| |

Systems in Triangular and Echelon Forms | |

| |

| |

| |

Gaussian Elimination | |

| |

| |

| |

Echelon Matrices, Row Canonical Form, Row Equivalence | |

| |

| |

| |

Gaussian Elimination, Matrix Formulation | |

| |

| |

| |

Matrix Equation of a System of Linear Equations | |

| |

| |

| |

Systems of Linear Equations and Linear Combinations of Vectors | |

| |

| |

| |

Homogeneous Systems of Linear Equations | |

| |

| |

| |

Elementary Matrices | |

| |

| |

| |

LU Decomposition | |

| |

| |

| |

Vector Spaces | |

| |

| |

| |

Introduction | |

| |

| |

| |

Vector Spaces | |

| |

| |

| |

Examples of Vector Spaces | |

| |

| |

| |

Linear Combinations, Spanning Sets | |

| |

| |

| |

Subspaces | |

| |

| |

| |

Linear Spans, Row Space of a Matrix | |

| |

| |

| |

Linear Dependence and Independence | |

| |

| |

| |

Basis and Dimension | |

| |

| |

| |

Application to Matrices, Rank of a Matrix | |

| |

| |

| |

Sums and Direct Sums | |

| |

| |

| |

Coordinates | |

| |

| |

| |

Linear Mappings | |

| |

| |

| |

Introduction | |

| |

| |

| |

Mappings, Functions | |

| |

| |

| |

Linear Mappings (Linear Transformations) | |

| |

| |

| |

Kernel and Image of a Linear Mapping | |

| |

| |

| |

Singular and Nonsingular Linear Mappings, Isomorphisms | |

| |

| |

| |

Operations with Linear Mappings | |

| |

| |

| |

Algebra A(V) of Linear Operators | |

| |

| |

| |

Linear Mappings and Matrices | |

| |

| |

| |

Introduction | |

| |

| |

| |

Matrix Representation of a Linear Operator | |

| |

| |

| |

Change of Basis | |

| |

| |

| |

Similarity | |

| |

| |

| |

Matrices, and General Linear Mappings | |

| |

| |

| |

Inner Product Spaces, Orthogonality | |

| |

| |

| |

Introduction | |

| |

| |

| |

Inner Product Spaces | |

| |

| |

| |

Examples of Inner Product Spaces | |

| |

| |

| |

Cauchy-Schwarz Inequality, Applications | |

| |

| |

| |

Orthogonality | |

| |

| |

| |

Orthogonal Sets and Bases | |

| |

| |

| |

Gram-Schmidt Orthogonalization Process | |

| |

| |

| |

Orthogonal and Positive Definite Matrices | |

| |

| |

| |

Complex Inner Product Spaces | |

| |

| |

| |

Normed Vector Spaces (Optional) | |

| |

| |

| |

Determinants | |

| |

| |

| |

Introduction | |

| |

| |

| |

Determinants of Orders 1 and 2 | |

| |

| |

| |

Determinants of Order 3 | |

| |

| |

| |

Permutations | |

| |

| |

| |

Determinants of Arbitrary Order | |

| |

| |

| |

Properties of Determinants | |

| |

| |

| |

Minors and Cofactors | |

| |

| |

| |

Evaluation of Determinants | |

| |

| |

| |

Classical Adjoint | |

| |

| |

| |

Applications to Linear Equations, Cramer's Rule | |

| |

| |

| |

Submatrices, Minors, Principal Minors | |

| |

| |

| |

Block Matrices and Determinants | |

| |

| |

| |

Determinants and Volume | |

| |

| |

| |

Determinant of a Linear Operator | |

| |

| |

| |

Multilinearity and Determinants | |

| |

| |

| |

Diagonalization: Eigenvalues and Eigenvectors | |

| |

| |

| |

Introduction | |

| |

| |

| |

Polynomials of Matrices | |

| |

| |

| |

Characteristic Polynomial, Cayley-Hamilton Theorem | |

| |

| |

| |

Diagonalization, Eigenvalues and Eigenvectors | |

| |

| |

| |

Computing Eigenvalues and Eigenvectors, Diagonalizing Matrices | |

| |

| |

| |

Diagonalizing Real Symmetric Matrices and Quadratic Forms | |

| |

| |

| |

Minimal Polynomial | |

| |

| |

| |

Characteristic and Minimal Polynomials of Block Matrices | |

| |

| |

| |

Canonical Forms | |

| |

| |

| |

Introduction | |

| |

| |

| |

Triangular Form | |

| |

| |

| |

Invariance | |

| |

| |

| |

Invariant Direct-Sum Decompositions | |

| |

| |

| |

Primary Decomposition | |

| |

| |

| |

Nilpotent Operators | |

| |

| |

| |

Jordan Canonical Form | |

| |

| |

| |

Cyclic Subspaces | |

| |

| |

| |

Rational Canonical Form | |

| |

| |

| |

Quotient Spaces | |

| |

| |

| |

Linear Functionals and the Dual Space | |

| |

| |

| |

Introduction | |

| |

| |

| |

Linear Functionals and the Dual Space | |

| |

| |

| |

Dual Basis | |

| |

| |

| |

Second Dual Space | |

| |

| |

| |

Annihilators | |

| |

| |

| |

Transpose of a Linear Mapping | |

| |

| |

| |

Bilinear, Quadratic, and Hermitian Forms | |

| |

| |

| |

Introduction | |

| |

| |

| |

Bilinear Forms | |

| |

| |

| |

Bilinear Forms and Matrices | |

| |

| |

| |

Alternating Bilinear Forms | |

| |

| |

| |

Symmetric Bilinear Forms, Quadratic Forms | |

| |

| |

| |

Real Symmetric Bilinear Forms, Law of Inertia | |

| |

| |

| |

Hermitian Forms | |

| |

| |

| |

Linear Operators on Inner Product Spaces | |

| |

| |

| |

Introduction | |

| |

| |

| |

Adjoint Operators | |

| |

| |

| |

Analogy Between A(V) and C, Special Linear Operators | |

| |

| |

| |

Self-Adjoint Operators | |

| |

| |

| |

Orthogonal and Unitary Operators | |

| |

| |

| |

Orthogonal and Unitary Matrices | |

| |

| |

| |

Change of Orthonormal Basis | |

| |

| |

| |

Positive Definite and Positive Operators | |

| |

| |

| |

Diagonalization and Canonical Forms in Inner Product Spaces | |

| |

| |

| |

Spectral Theorem | |

| |

| |

| |

Multilinear Products | |

| |

| |

| |

Algebraic Structures | |

| |

| |

| |

Polynomials over a Field | |

| |

| |

| |

Odds and Ends | |

| |

| |

Index | |