Linear Algebra

• Algebraic Structures
  • Group
    • Abelian Group
  • Ring
  • Field
• Vector Space and Geometry
  • Vector Space
    • Euclidean Vector Space
      • Vector Space
      • Vector Subspace
      • Linear Dependence and Independence
      • Basis of a Vector Space
      • Inner Product and Orthogonality of Vectors
  • Affine Sets in Euclidean Vector Space
    • Line
    • Plane
  • Orthogonal Projections
• Linear Transform
  • Linear Transform in Vector Space
    • Linear Operator
    • Space of Linear Transforms
      • As a Vector Space over Addition? Always.
    • Composition of Linear Transforms
      • As a Vector Space over Composition? Not Guaranteed.
    • Examples
      • Scalar Multiplication as a Linear Transform
      • Matrix-Vector Multiplication as Linear Transform
      • Differentiation as a Linear Transform
      • Integration as a Linear Transform
  • Matrix-vector multiplication
  • Matrix-matrix multiplication
  • Independence, Rank, Inverse Mapping, Basis and Fundamental Subspaces
    • Independence
    • Rank
    • Inverse Mapping
    • Basis
    • Fundamental Subspaces
  • Orthogonality
    • Orthogonal vectors
    • Orthonormal vectors
    • Matrix with orthonormal columns
    • Projection matrices
    • Orthogonal matrices
    • Orthonormal basis
    • Orthogonal subspace
• Matrix Factorisation
  • A=CR
  • LU Factorisation
  • LDLT Factorisation
  • Cholesky Factorisation
  • Gram-Schmidt Orgthogonalisation
  • Eigendecomposition
    • Real and Complex Eigenvalues
    • Matrix power
    • Trace and Determinant
    • Similar Matrices
    • Properties
    • Special case: Symmetric Real Matrices
      • Positive Definite Matrices
        • Tests for positive definiteness
      • Positive Semi-definite Matrices
  • Singular Value Decomposition
    • Computing singular values and vectors
    • Eckart-Young: Best rank k approximation
• Vector and Matrix Norms
  • Vector Norms
    • l1 Norm
    • l2 Norm
    • l infinity Norm
  • Matrix Norms
    • Spectral Norm
    • Frobenius Norm
    • Nuclear Norm
• Applied Linear Algebra
  • Principle Component Analysis
  • Matrix Identities
  • Einstein Summation
  • Resources
• Random Matrix Theory and Randomized Linear Algebra
  • Resources