Eulerly

Linear algebra you can touch

Free interactive explorations from the course Applied Linear Algebra: The Math Behind AI and Beyond. Drag, turn the dials, and watch the maths move. Every page shows the matching line of NumPy.

Module 2 · Vectors: The Toolkit

Lesson 2.2 · Length and norms

Norm explorer

Drag a vector and compare three ways to measure it: the drone (L2), the rider (L1) and the crane (L∞), with their circle, diamond and square.

Lesson 2.3 · Linear combinations

Linear combination dials

Turn the weights m and n, leave a dot at every setting, fill the plane, and hit targets. Drag a and b to discover when the span collapses to a line.

Module 3 · Dot Products, Angles and Similarity

Lesson 3.3 · Cosine similarity

Cosine similarity dial

Drag two arrows and watch a dial swing from −1 (opposite) through 0 (unrelated) to +1 (alike). Stretch an arrow and see that the needle does not move.

Lesson 3.4 · Projections

Projection explorer

Cast the shadow of b on the line of a: scalar projection, vector projection, and a remainder that always meets a at a right angle. Drop b behind a for a backward shadow.

Module 5 · Matrices as Transformations

Lessons 5.1–5.4 · Matrices as transformations

Transformation playground

Edit a 2×2 matrix and watch the grid and the unit square move. See det T as an area, spot a flip or a squash, then chain presets and see that the order changes the result.

Modules 6 and 7 · Systems of Linear Equations, Row Reduction and Rank

Hyperplanes

Hyperplane classifier

One equation, w · x + b = 0, splits the plane. Drag w and the points, read each score and signed distance, and double w and b to see the line stay put.

Elimination

Elimination stepper

Step through row operations on any system up to 4×4: pivots, row echelon form, rank, back substitution and RREF. Compare partial pivoting on and off.