Lab: Vector Playground
One number tells you: line or plane?
What you will do
- The determinant decides independence
- A zero determinant collapses the span
- The angle between two vectors
- Projection divides by the square
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
Lab: Matrix Transformation Visualizer
Drag a matrix, watch the plane move
What you will do
- The determinant as area
- A negative determinant flips
- The determinant of a composition
Lab: Linear System Solver
One check tells you: one, none, or infinitely many?
What you will do
- The determinant decides the unique case
- A rank gap means no solution
- A free variable means infinitely many
- The unique solution of a 3×3
Lab: Eigenvalue Explorer
Which directions does a matrix leave alone?
What you will do
- A symmetric matrix has real eigenvalues
- Its eigenvectors are orthogonal
- The trace is the sum of the eigenvalues
- The determinant is the product of the eigenvalues
- A rotation has no real eigen-direction
Lab: Orthogonality and Projections
How do you drop a vector straight down onto a subspace?
What you will do
- Gram-Schmidt returns perpendicular vectors
- Each basis vector has unit length
- Projection drops the target straight down
- The residual is orthogonal to the subspace
- Least squares fits collinear points exactly
- An exact fit has zero residual
Lab: Fourier Series Builder
How can a stack of pure sine waves rebuild a sharp square wave?
What you will do
- The fundamental amplitude is 4/π
- Even harmonics vanish
- Different harmonics are orthogonal
- Each harmonic's square-norm is T/2
- More terms means less error
- The Gibbs overshoot never goes away
Lab: Matrix Decomposition Explorer
Take a matrix apart, keep only what matters
What you will do
- The rank is the count of non-zero σi
- Energy captured by the top k
- Reconstruction error falls, then vanishes
Lab: Gradient Descent Visualizer
Walk downhill on a quadratic
What you will do
- The gradient at a point
- One step lands lower
- Where descent stops converging
- Just past the threshold: it diverges
Lab: Numerical Experiments
When does a matrix amplify the errors you feed it?
What you will do
- A perfectly-conditioned matrix has κ = 1
- A stretched matrix is ill-conditioned
- A diagonally-dominant system has small ρ
- ρ < 1 means the iteration converges
- Without dominance, ρ > 1 and it diverges
- A near-singular system amplifies error by κ
Lab: PCA Explorer
Keep the few directions your data actually uses
What you will do
- Variance concentrates in the top PCs
- How many PCs reach 95%
- The dimensionality-reduction ratio
Project: ML with Linear Algebra
Three ML primitives, one arithmetic
What you will do
- Regression: the slope from scratch
- Factorize: the energy in the top k
- Forward pass: the output
Project: Data Analysis Pipeline
Reduce first, then model
What you will do
- Reduce: how many PCs for 95%
- Reduce: the reduction ratio
- Model: how much variance the fit explains
- Model: the fitted slope
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.