Algebra

Algebra studies linear structure: vectors built by linear combination, maps that preserve linear combinations and the coordinates that turn both into lists and matrices. The strand works over a scalar field $F$, usually $\mathbb R$ for geometric examples and $\mathbb C$ when complex roots or eigenvalues are needed.

Chapters

1. Complex numbers: arithmetic and polar form in $\mathbb C$, polynomial roots and roots as modes of discrete and continuous systems.
2. Vector spaces: one setting for $F^n$, polynomial spaces $P_n(F)$ and matrix spaces $M_{m\times n}(F)$, developed through subspaces, span, independence, bases and coordinates and ending with interpolation.
3. Linear transformations: maps that preserve linear combinations, read through their geometry, kernel and image and, in finite dimensions, as matrices once bases are chosen.
4. Eigenvalues: invariant directions and diagonalisation, including when a matrix is not diagonalisable over the chosen field, applied to matrix powers, linear ODE systems and Markov chains.

Links to Calculus

• Polynomial spaces: interpolation and Taylor models are both coordinate problems in a polynomial space.
• Kernels: the kernel of a linear map describes every solution of a homogeneous equation and every solution of a forced equation once one particular solution is known. This covers matrix equations and linear ODEs alike.
• Characteristic roots: complex characteristic roots describe the growth, decay and oscillation of recurrences and of the differential equations in Calculus.

See Core for how the two strands fit together.