Concept

Shear — where it appears

A map that slides each line parallel to a fixed direction by an amount depending on its distance, leaving area unchanged.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

algebra · linear maps

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 · eigenvectors

Euclid proves it without moving anything

The rearrangement proof cuts and slides. Euclid's does neither — it shows that a square and a rectangle are each exactly twice the same triangle, seen from opposite sides, and that is harder to hold in the head for a reason worth understanding.

geometry · pythagoras

Named alongside it

The objects these essays reach for when they reach for this one.

Area Basis Determinant Matrix Altitude Characteristic polynomial Congruence Diagonalisation Dissection Eigenvalue Eigenvector Hypotenuse

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