Linear algebra
The theory of vector spaces and the maps between them. The corpus holds one account — hefferon-linear-algebra, a first course proving everything from one semester of calculus.
Named in synthesis as the largest hole outright, on the grounds that it “sits under every text here and is the subject of none.” That description was accurate and worth keeping in view now that it is filled, because the dependency runs one way: every other subject in this spoke consumes linear algebra and none of them supply it.
The arc
- Linear systems — Gauss’s method and reduced echelon form. The concrete anchor everything abstract is later shown to be about.
- Vector spaces — subspaces, span, linear independence, basis, dimension.
- Linear maps and matrices — homomorphism and isomorphism, matrix representation, change of basis, projection. The chapter where a matrix stops being an array of numbers and becomes a map in a chosen basis.
- Determinants — Laplace expansion, and the geometric reading as signed volume scaling.
- Eigenvectors and eigenvalues — similarity, diagonalization, Jordan form.
What consumes it here
- vector-calculus — the derivative of a several-variable map is a linear map, and the Jacobian is its matrix. Change of variables in a multiple integral is a determinant.
- fourier-analysis — orthogonality, projection onto a basis, and linear systems are the chapter-3 vocabulary applied to function spaces. thibos-fourier-analysis uses the whole of it without naming the source.
- probability-theory — covariance is a bilinear form; the corpus’s counting arguments run through symmetric group actions (mcmullen-probability-theory).
- Outside the spoke, it is the machinery under most of the hub’s applied numerics.
On rigour
This subject is where the corpus’s rigour axis stops being a trade-off. corral-vector-calculus gives up proofs to stay reachable and says so; hefferon-linear-algebra proves everything and still asks only one calculus course, by spending pages on motivation instead of assuming mathematical maturity as an entry condition. That is an argument that the axis this spoke has been recording is partly an artefact of length.
Related
hefferon-linear-algebra · jim-hefferon · vector-calculus · fourier-analysis · mathematics