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Concept

Cauchy schwarz

The inequality saying that an inner product is never larger in size than the product of the two lengths. Equality holds exactly when one vector is a multiple of the other, and the bound is what makes the ratio of a product to two lengths a cosine, so that an angle can be defined in a space with no picture.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The parabola that proves |a·b| ≤ |a||b|. The squared length of a − t b plotted against t. It is a parabola opening upward whose least value is 7.118; that this is never negative is exactly the Cauchy–Schwarz inequality.

The square that cannot be negative

Cauchy–Schwarz is the load-bearing inequality of the whole subject and is nearly always stated without proof. It is one line away from a fact nobody would argue with, and the line is a parabola with no room to cross the axis.

algebra · Inner product One subtraction clears a direction. Gram–Schmidt on two planar vectors, in 3 panels: the pair as given, the shadow of the second on the first, and the perpendicular pair that is left when the shadow is removed.

One subtraction clears a direction

A basis is a set of directions to measure along, and most bases are awkward because the directions get in each other's way. Removing one shadow at a time turns any basis into one where every coordinate is a shadow and nothing interferes.

algebra · Inner product Perpendicular to both, and as long as their parallelogram. The vectors (2, 0.4, 0.2) and (0.6, 1.8, 0.3), the parallelogram they span, and their cross product (−0.24, −0.48, 3.36), drawn perpendicular to both with a length of 3.403, which is the parallelogram's area.

A plane disguised as an arrow

The cross product of two arrows is an arrow perpendicular to both, as long as the area of their parallelogram. Reflect everything in a mirror and it points the wrong way, because it was never an arrow: it is a plane, written as the one direction a plane in three dimensions leaves over.

algebra · Inner product Parity and majority on the cube, and the edges between true and false. Two cubes of three-letter assignments, parity with all twelve edges joining a true and a false corner and majority with 6; Khrapchenko's bound at its best 9.00 against an exact size of 10 and 4.00 against an exact size of 5.

The formula that cannot share

A circuit computes the parity of n letters with a chain of n − 1 exclusive-or gates. A formula in and, or and not, which may use each result only once, needs n² letters at least — a bound Khrapchenko proved by counting the cube's edges between true and false — and exhaustive search over every formula of four letters finds parity sitting exactly on it.

logic · Truth functions

Named alongside it

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

Inner productOrthogonalityDot productPositive definiteProjectionApproximationBasisBoolean formulaCircuit complexityCross productDeterminantDiscriminant

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