**Introduction to linear transformations**

- 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
- More on the determinant: multiplicativity, explicit formula, explicit form of inverse
- 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)

Basis changes analogies: 2nd order homogenous linear DEs when you get complex solutions; Integral substitution, integrate r^2 e^(-r^2) dr from 0 to infinity; generic substitutions. Number of free variables staying the same ("n variables for n equations", row rank = column rank)

**Tracey stuff**

- Trace and its links to determinant and eigenvalues
- 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)

**Decompositions**

- SVD, polar decomposition, normal matrices; re-look at transposes, FTLA
- Triangular matrices; Schur, Cholesky, QR, LU decompositions

See also Algorithms for the numerical details of such decompositions, other numerical linear algebra.

**Euclidean geometry**

- Orthogonal complements and cross products
- Weirdness with planes
- Projections
- Dot products and the greedy algorithm
- Abstract Euclidean geometry

**Tensors**

- Introduction to tensors and index notation
- Covectors, conjugates, and the metric tensor
- Higher-order tensors -- intro, Le Cevita tensor (determinants as multilinear forms), partial trace
- Transformation of tensors
- Tensor products, tensor derivatives
- Tensors in mechanics

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