[home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff]
	Mathematics-Online course:
	Linear Algebra

Table of Contents:

• Basic Structures
  • Groups and Fields
    • Group
    • Subgroup
    • Operation Tables
    • Permutations
    • Cycle Notation of Permutations
    • Transposition and Sign of Permutations
    • Modulus
    • Residue Classes modulo n
    • Fields
    • Prime Fields
    • Galois Fields
    • Tournaments in Groups
  • Vector Spaces
    • Vector Space
    • Vector Space of n-Tuples
    • Subspace
    • Linear Combinations
    • Linear Hull
    • Convex Combinations
    • Convex Hull
  • Scalar Product and Norm
    • Properties of Scalar Products
    • Scalar Product
    • Cauchy-Schwarz Inequality
    • Norm
    • Scalar Product - Norm
    • Maximum Norm
    • Scalar Product of Real Vectors
    • Scalar Product of Complex Vectors
    • Angles
    • Pythagorean Theorem
    • Pythagorean Triple
  • Bases
    • Linear Independence
    • Basis
    • Dimension
    • Orthogonal Basis
    • Orthogonal Projection
    • Gram-Schmidt Method
• Matrices
  • Linear Maps
    • Linear Maps
    • Composition of Linear Maps
    • Matrix
    • The Matrix of a Linear Map
    • Coordinate Transformation under Change of Basis
    • Image and Kernel
    • The dimensions of Image and Kernel
    • Inverse Map
  • Matrix Operations
    • The Vector Space of Matrices
    • Matrix Multiplication
    • Strassen's Algorithm
    • Commutator
    • Inverse Matrix
    • Transpose of and adjoint Matrix
    • Trace
    • Rank of a Matrix
    • Norm of a Matrix
    • Maximum Absolute Row Sum Norm
  • Special Matrices
    • Especially Structured Matrices
    • Orthogonal and Unitary Matrices
    • Hadamard Matrices
    • Fourier Matrices
    • Isometry of Orthogonal and Unitary Matrices
    • Cyclic Matrices
    • Stochastic Matrices
  • Determinants
    • Determinant as Antisymmetric Multilinear Form
    • 2-by-2 Determinant
    • Sarrus Scheme
    • Oriented Volume
    • Determinant Rules
    • Determinants of Special Matrices
    • Transformation of Determinants
    • Expansion of Determinants
    • Plane through 3 Points
    • Basis Test
    • Vandermonde Determinant
• Linear Systems of Equations
  • Classification and General Structure
    • Linear System of Equations
    • Electric Circuit
    • Solvability of a linear system of equations
    • Solution of Linear Systems of Equations and Matrix Inversion
    • Determinant and Solvability of a linear system of equations
  • Direct Methods
    • Cramer's Rule
    • Substituting Backwards
    • Gauss Transformations
    • Gauss Elimination for an invertible Matrix
    • Gauss Elimination to Echelon Form
    • Solution of a LSE in Echelon Form
    • Rank and Solvability of SLE
  • Approximation Problems
    • Best Fitting Line
    • Approximative Solution of Over-determined Linear Systems of Equations
    • Normal Equation
    • Tomography
• Normal Forms
  • Eigenvalues and Eigenvectors
    • Eigenvalue and Eigenvector
    • Similarity Transformation
    • Characteristic Polynomial
    • Calculation of Eigenvalues and Eigenvectors
    • Algebraic and Geometric Multiplicity
    • Sum and Product of Eigenvalues
    • Rational Functions of Matrices
    • MATLAB - Eigenvalues and Eigenvectors
  • Diagonalisation
    • Basis of Eigenvectors
    • Diagonalisation of Cyclic Matrices
    • Powers of Matrices
    • Fibonacci Sequence
    • Normal Matrices
    • Unitary Diagonalisation of Normal Matrices
    • Diagonal From of Hermitian Matrices
    • Positive Definite Matrices
    • Rayleigh Quotient
  • Jordan Normal Form
    • Triangular Form
    • Generalized Eigenvectors
    • Cyclic Bases of Generalized Eigenspaces
    • Jordan Form
    • Jordan Form of (4x4)-Matrices
    • Powers of Matrices
  • Singular Value Decomposition
    • Singular Value Decomposition
    • MATLAB - Singular Value Decomposition
    • Pseudo-Inverse
    • Topography
    • Moore-Penrose Conditions
• Analytic Geometry
  • Orthogonal Groups
    • Classification and Special Transformations
    • Reflection
    • Rotation
    • Rotation Axis and Angle
    • Decomposition
    • Rotation Matrix
  • Quadrics
    • Quadratic Form
    • Quadric
    • Simplistic Classification of Quadrics
    • Principal Axis Transformation
    • Cone Sections
    • Euclidean Normal Form of two-dimensional Quadrics
    • Euclidean Normal Form of three-dimensional Quadrics