Matpic
mathematics, in pictures
Generator

linear-map

A generator in the algebra library, called 15 times across 4 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

linear-map is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

A linear map redrawing the planeThe integer grid before and after a linear transformation; the shaded unit square becomes a parallelogram whose area is the determinant.beforeafter · area × 2.50210.51.5

show: "columns"

A matrix's columns are where the basis vectors landThe first column gives the destination of the vector pointing one unit east; the second gives the destination of the vector pointing one unit north.eastnorth(2, 0.5)(1, 1.5)210.51.5column 1column 2

show: "zoo"

Four linear mapsA shear, a rotation, a squeeze and a singular map, each applied to the same grid and unit square.sheardet = 1 · area unchangedrotationdet = 1 · area unchangedsqueezedet = 0.99 · area unchangedcollapsedet = 0 · area zero

show: "eigen"

The directions the map leaves aloneUnit vectors and their images under the map. On the two marked lines the image points the same way as the original, stretched by 3.00 and 1.00.λ = 3.00λ = 1.00[2, 1, 1, 2]

What it checks while it draws

Collected by running the family and listening to lib/verify.js, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Algebra

A matrix is a picture of what happens to the grid

Four numbers in a box is not an object anyone has intuitions about. The same four numbers, shown as an instruction for redrawing the plane, are.

 Topology

Something always stays put

Stir a cup of coffee however violently and let it settle. Some molecule is exactly where it started. Crumple a map and drop it on the region it depicts, and one point lies over the place it names.

 Algebra

The directions a map leaves alone

Almost every arrow comes out of a transformation pointing somewhere else. A few come out pointing exactly where they went in, only longer or shorter. Those few decide nearly everything the map does.

 Algebra

The dot product is a shadow

Multiply the matching coordinates and add them up. That rule explains nothing, and it hides the fact that the answer is a length — how far one arrow reaches along another, times how long that other one is.

The whole library · What the figures prove