Linear Algebra

The fundamental ideas of linear algebra: vector spaces, linear maps, determinants, inner products, and the spectral theorem — developed rigorously with an emphasis on geometric intuition and the distinction between operators and their matrix representations.

Work in Progress

This resource is actively under development and edits occur frequently.

Content is being added chapter by chapter. Stay tuned for updates.

This resource covers the fundamental ideas of linear algebra — not a survey, but the core results developed carefully, with full proofs and the geometry made explicit at every step. The two central distinctions that run throughout:

Operators vs matrices — an operator is a geometric object; a matrix is its coordinate representation in a chosen basis. Similar matrices represent the same operator. This distinction matters everywhere, from eigenvalues to the spectral theorem.
Geometry first, coordinates second — every concept is introduced abstractly and then computed in coordinates. The abstract version reveals what is really going on; coordinates make it usable.

The topics progress from the foundations (vector spaces, linear maps, determinants) through the structure theory (eigenvalues, inner products, orthogonality, the spectral theorem) to two major results that pull everything together: the singular value decomposition and the stability theory of linear dynamical systems.

How to Read This

The chapters are designed to be read in order — later chapters use results from earlier ones freely. That said, there are natural stopping points depending on what you are after:

Foundations only (Ch. 1–7): vector spaces, linear maps, matrices, change of basis, linear systems, determinants. This is self-contained and covers the core algebraic machinery.
Through spectral theory (Ch. 1–12): adds eigenvalues, inner products, orthogonality, and the spectral theorem. This is the natural endpoint for most purposes.
Full text (Ch. 1–16): continues to the SVD, least squares, and linear dynamical systems/Markov chains.

Statements and examples are readable on their own, and the proofs are there for those who want to see why the results are true. Reading a proof once, even if you cannot yet reconstruct it, is worth doing.

Coming Soon

Problem sets will be added to each chapter — exercises ranging from routine calculations to proofs, organized by difficulty.

Diagrams and animations will accompany the geometric content throughout: subspace projections, eigenvalue geometry, the SVD ellipsoid, Markov chain convergence, and more.

Interactive Tools

The Coordinate Maps tool visualizes how linear maps (and general maps) act on \mathbb{R}^2 and \mathbb{R}^3: enter a matrix or formula, and see how it transforms the grid, the unit circle, and the unit square in real time. It also displays the Jacobian at any point. This is useful throughout — for building intuition about projections, rotations, shears, eigenvalue stretching, and the SVD ellipsoid.