Linear Algebra

About these notes

Linear algebra provides a language for simultaneous equations, geometry, approximation and dynamics. These introductory chapters move from solving equations to understanding vector spaces and the linear maps between them.

Prerequisites and conventions

Review algebraic equations and basic function notation. Vectors are columns, scalars are real unless another field \(\mathbb F\) is specified, and matrix multiplication represents composition with the rightmost map acting first. Proofs use elementary logic; Discrete Mathematics (planned) owns the broader proof-methods outline.

Written chapter map

1. Linear systems and Gaussian elimination: consistency, pivots, rank and free variables.
2. Vector spaces, span, independence and basis: coordinates, dimension and subspaces.
3. Linear transformations and matrix representations: kernel, image, rank-nullity and change of basis.

Examples and answer checks accompany each chapter. Try the exercises before opening the answers.

Planned chapters

The following are not yet written: determinants and invertibility; eigenvalues, eigenvectors and diagonalization; inner products, orthogonality, projection and least squares; LU, QR and singular value decompositions; the spectral theorem for real symmetric and complex Hermitian matrices.

This book owns the mathematical structure of these ideas. Floating-point implementations, conditioning and numerical stability belong to Numerical Analysis (planned). Applications to differential systems belong to ODEs (planned).

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