1. Matrix machine with adjustable collapse
Use the sliders to define a matrix. Watch how a grid of input points transforms. When the transformed grid collapses toward a line, information is being lost.
2. Column space: reachable targets
The two columns of a matrix are output directions. Move the target point. The page checks whether the target is reachable for the rank-one matrix below.
A = [[1, 2], [2, 4]]
3. Null space: invisible directions
For the same rank-one matrix, moving along the null direction changes the input but keeps the output unchanged.
Here x₀ = (1,0) and v = (-2,1). Since Av = 0, all inputs x₀ + tv have the same output.
4. Rank-nullity calculator
Choose the input and output dimensions and an intended rank. The calculator shows how many input directions must disappear.
5. Near information loss
The matrix Aε = [[1,1],[1,1+ε]] is invertible when ε ≠ 0, but as ε becomes small its columns become almost dependent.
A matrix can have full rank and still be dangerous numerically. Near information loss creates large sensitivity.
6. Image compression preview
This toy image is a matrix. A rank-k approximation keeps only k directions. Smaller k means more compression and more information loss.