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. Basis changes; commuting matrices
  8. Eigen-everything, multiplicity
  9. Invariant and generalised eigenspaces; Jordan normal form
  10. All matrices can be diagonalised over R[X]/(X^n)
  11. 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)
  5. That silly formal determinant
Decompositions
  1. SVD, polar decomposition, normal matrices; re-look at transposes, FTLA
  2. Triangular matrices; Schur, Cholesky, QR, LU decompositions
Reference
  • Computational techniques (proofs regarding direct formulae for determinants, inverses -- what is those adjoint/minor/cofactor stuff anyway?)
  • The geometry of $\mathbb{R}^n$ -- cross product, planes, projections, 3D rotations, etc.
Resources
  • The Essence of Linear Algebra youtube video series (3Blue1Brown)
  • Mathview (for tensors)

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