**Introductory**

- Introduction to linear transformations
- Linear sums and independence, span, and linear bases
- Why matrices?
- Composition of linear transformations: matrix multiplication
- Inverses, determinants, column spaces, non-square matrices
- Null, row spaces, transpose, fundamental theorem of linear algebra
- Basis changes; commuting matrices
- Eigen-everything, multiplicity
- Invariant and generalised eigenspaces; Jordan normal form
- All matrices can be diagonalised over R[X]/(X^n)
- An introduction to forms

**Tracey stuff**

- Trace and its links to determinant and eigenvalues
- Some interesting theorems in linear algebra; operator norms
- Orthogonal group, indefinite orthogonal group, orthochronous stuff
- Geometry, positive definiteness, and Sylvester's law of inertia
- Complex linear algebra; the spectral theorem
- The Hilbert space

**Insight into cross products series**

- Quaternions introduction: Part I
- Quarternions introduction: Part II
- Octonions and beyond
- Exterior products (aka: a primer on tensors)
- That silly formal determinant

**Decompositions**

- SVD, polar decomposition, normal matrices; re-look at transposes, FTLA
- 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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