Linear algebra

Introduction to linear transformations

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. More on the determinant: multiplicativity, explicit formula, explicit form of inverse
7. ★★★★★ Null, row spaces, transpose, fundamental theorem of linear algebra
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)

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

1. Trace and its links to determinant and eigenvalues
2. ★★★☆☆ Orthogonal group, indefinite orthogonal group, orthochronous stuff
3. ★★★★☆ Geometry, positive definiteness, and Sylvester's law of inertia
4. Complex linear algebra; the spectral theorem
5. 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

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

Euclidean geometry

1. Orthogonal complements and cross products
2. Weirdness with planes
3. Projections
4. Dot products and the greedy algorithm
5. Abstract Euclidean geometry

Tensors

1. ★★★☆☆ Introduction to tensors and index notation
2. ★★★☆☆ Covectors, conjugates, and the metric tensor
3. Higher-order tensors -- intro, Le Cevita tensor (determinants as multilinear forms), partial trace
4. Transformation of tensors
5. Tensor products, tensor derivatives
6. Tensors in mechanics