Linear algebra
Vectors, matrices, eigenvalues, inner products, the spectral theorem.
Linear algebra is the language of transformations: things that combine inputs additively and respect scaling. Once a phenomenon is linear — and an astonishing fraction of physics is linear, at least near equilibrium — its full structure can be captured in finite-dimensional terms by vectors and matrices, or in infinite-dimensional terms by operators on function spaces.
This chapter is the working refresher. It is here because the rest of the bookshelf invokes linear-algebra concepts repeatedly — eigenvalues of a Jacobian in the ODE phase plane, eigenfunctions of the wave operator in PDEs, modes of a cavity in Helmholtz, energy eigenstates in Schrödinger, the orthogonal basis underlying every Fourier expansion — and assumes the reader has the tools in hand. We collect those tools here.
The chapter is built as a gentle ramp:
- 4.1 What is linear algebra? — vectors as columns of numbers, matrices as transformations of the plane, the geometric picture before any algebra.
- 4.2 Vectors and matrices — the working operations: addition, scalar multiplication, matrix–vector and matrix–matrix product, the transpose, the inverse and the determinant.
- 4.3 Linear systems: solving Ax = b — Gaussian elimination, the three things that can happen (unique solution, no solution, infinitely many), and what the geometry says.
- 4.4 Eigenvalues and eigenvectors — the special directions a matrix only scales rather than rotates, the characteristic polynomial, power iteration, and why this is what the ODE and PDE chapters were already using.
- 4.5 Inner products and orthogonality — geometry on a vector space, projections, the Gram–Schmidt procedure, orthonormal bases.
- 4.6 Self-adjoint operators and the spectral theorem — the deep theorem that ties everything together and underwrites separation of variables in PDEs.
If you have not done linear algebra in a while, this chapter is the audience it was written for. Each lesson reintroduces its idea from the picture down before any algebra is required.