Matpic
mathematics, in pictures
Concept

Dimension

The number of independent directions in a space, and so the number of coordinates a point in it needs.

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

before — the line that will vanishafter — all of it on one linethe matrix sends every point of one line — the direction (-0.447, 0.894) — to the origin, and everything else onto a singleline through itone dimension is lost and one survives; that they add to the two the plane started with is the rank-nullity theorem, and it isthe whole content of the picture

What a map throws away

A linear map redraws the grid, and the determinant measures how much it stretches area. When that measurement comes out zero the map has flattened the plane onto a line — and the question worth asking is not how much was lost but how much survived, because the two always add to what there was.

algebra · linear maps 124312.6513.00a floor 12 by 4 makes a diagonal of √(12² + 4²) = 12.649that diagonal and the height of 3 are the legs of a second right triangle, whose hypotenuse is√(12.649² + 3²) = 13.000 = √(144 + 16 + 9)

Two right angles and the diagonal of a box

The theorem applied once gives the diagonal of a floor. Applied again, standing on the first result, it gives the diagonal of the room — and the pattern does not stop at three, which is where a fact about triangles quietly becomes the definition of distance.

geometry · pythagoras

Named alongside it

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

BasisDe gua theoremDeterminantDistanceHypercubeImageKernelLinearityMatrixNullityOrthogonalityProjection

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