Here’s the current “table of contents”

Linear Algebra

• Why linear algebra – an overview
• Vector Spaces
  • Basic Definitions and Examples
  • Linear Transformations and Matrices
  • Change of Basis
  • Rank and Solutions of Linear Systems
  • Sums, Intersections and Projections
  • Realification and Complexification
  • The Dual Vector Space
• Linear Operators
  • Determinants
  • Eigenvectors, Eigenvalues and Diagonalisation
  • The Trace
  • Diagonalisable Linear Operators
  • The Cayley-Hamilton Theorem
  • The Jordan Normal Form
  • The Real Jordan Normal Form
• Inner Product Spaces
  • Definitions and Examples
  • Low Dimensional Inner Product Spaces
  • Classification of Inner Product Spaces
  • Norms
  • Gram-Schmidt Orthogonalisation
  • Self-Adjoint, Unitary and Orthogonal Operators
  • Polar and Singular Value Decompositions
  • The Riesz Lemma
• Dirac Notation
  • Outer Products in Dirac Notation
  • Simultaneously Diagonalisable Operators
• Multilinear Algebra
  • Definition and Construction of the Tensor Product
  • Dimension and Bases
  • Isomorphisms
  • The Tensor Algebra
  • Component Representation of Tensors
  • Symmetric Tensors and the Symmetric Algebra
  • Skew-Symmetric Tensors and the Exterior Algebra
  • Tensor Symmetries in Coordinate Representation
  • The Determinant Revisited
  • The Hodge Dual