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 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as positive definite — the same set of essays touches all of them, so they are one junction rather than several.

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 asserted. 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

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

Named alongside it

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

Dot product Inner product Orthogonality Positive definite Projection Approximation Basis Discriminant Gram schmidt Inequality Least squares Linear independence

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