How to read these. Each page states which of its numbers are computed, which come from a published figure, and which are illustrative. Several were generated with AI assistance and say so on the page. They are teaching instruments, not measurements of real hardware.
Labs
Demos
Vector Playground
From lesson What is Linear Algebra?
Drag two vectors and watch their sum, dot product, angle and projection update live; the determinant decides whether their span is a line or the whole plane.
Matrix as a transformation of the plane
From lesson Linear Transformations
Drag a vector and watch the basis and the determinant follow. Eigenvectors emerge as the directions that do not turn.
Eigenvalue Explorer
From lesson The Eigenvalue Problem
Set a 2×2 matrix and watch its characteristic polynomial, eigenvalues and eigenvectors compute live; rotate a unit vector and see its image land parallel to itself only along an eigen-line, while a negative discriminant means a complex pair and no real eigen-direction.
Linear System Solver
From lesson Setting Up Linear Systems
Type a 2×2 or 3×3 system A x = b and watch Gaussian elimination with partial pivoting run to row echelon form, the determinant appear as the product of the pivots, and the system classified as one solution, none, or infinitely many — with the two lines drawn for the 2×2 case.
SVD and low-rank approximation
From lesson Singular Value Decomposition
Move a rank slider and watch an image rebuild itself one singular value at a time, with the retained energy computed at every rank.
Orthogonality & Projections
From lesson Orthogonal Vectors and Projections
Orthonormalize two vectors with Gram–Schmidt, project a vector onto the span with its orthogonal residual, and fit a line by least squares — every basis vector, projection, residual and coefficient computed from the normal equations, with the residual shown orthogonal to the column space.
Fourier Series Builder
From lesson Inner Products
Rebuild a periodic waveform (square, sawtooth, triangle, pulse) from its Fourier harmonics, with each coefficient computed as an inner product of the function with a sine or cosine — the series as an orthogonal projection. Watch the RMS error fall as terms are added, the orthogonality table sit at zero off the diagonal, and the ~9% Gibbs overshoot at a jump narrow but never shrink.
Numerical Experiments
From lesson Floating Point and Conditioning
Set a 2×2 or 3×3 matrix and watch the numerics: the condition number κ = σmax/σmin computed by power and inverse-power iteration on AᵀA, the image ellipse whose axis ratio is κ, Jacobi and Gauss–Seidel with their convergence trace and the spectral radius that decides convergence, and how ill-conditioning amplifies a perturbation of b within the κ bound.