Linear algebra

Introductory
  1. Introduction to linear transformations
  2. Linear sums and independence, span, and linear bases
  3. Why matrices?
  4. Composition of linear transformations: matrix multiplication
  5. Inverses, determinants, column spaces, non-square matrices
  6. Null, row spaces, transpose, fundamental theorem of linear algebra
  7. Importance of the multiplicativity of the determinant
  8. Basis changes; commuting matrices
  9. Eigen-everything, multiplicity
  10. Invariant and generalised eigenspaces; Jordan normal form
  11. All matrices can be diagonalised over R[X]/(X^n)
  12. An introduction to forms
Tracey stuff
  1. Trace and its links to determinant and eigenvalues
  2. Some interesting theorems in linear algebra; operator norms
  3. Orthogonal group, indefinite orthogonal group, orthochronous stuff
  4. Geometry, positive definiteness, and Sylvester's law of inertia
  5. Complex linear algebra; the spectral theorem
  6. The Hilbert space
Insight into cross products series
  1. Quaternions introduction: Part I
  2. Quarternions introduction: Part II
  3. Octonions and beyond
  4. Exterior products (aka: a primer on tensors)
Decompositions
  1. SVD, polar decomposition, normal matrices; re-look at transposes, FTLA
  2. Triangular matrices; Schur, Cholesky, QR, LU decompositions
Euclidean geometry
cross product, planes, projections, 3D rotations, etc. Dot product and greedy algorithm.
abstract Euclidean geometry

See also Computer Science for numerical linear algebra.

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