Concept

Linear dependence

A relation among vectors in which a non-trivial combination of them adds to zero. Any list longer than the dimension of the space is dependent, which is how a counting of dimensions proves that some polynomial equation must exist.

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

Every point of a hull, as a mixture of three of 11 points. A scatter of points with its convex hull outlined, and several interior points each shown inside a triangle of three of the scattered points, found by trying every triple.

Three points, however many there are

A point inside the hull of a thousand points is inside the hull of three of them. Any four points split into two groups whose hulls meet. And a family of convex sets, every three of which have a common point, has one common to all — three, in each case, being one more than the dimension.

analysis · Convexity

Seven powers of √2 + ∛3 in a space of six. A table of the powers 1 to (√2 + ∛3)⁶ as coordinate vectors over a six-element basis, with the coefficients of the dependency among them: the minimal polynomial x⁶ − 6x⁴ − 6x³ + 12x² − 36x + 1.

Seven powers in a space of six

Is √2 + ∛3 a root of some polynomial with whole-number coefficients? It lives in a field of dimension six, so its first seven powers are seven vectors in a six-dimensional space and must be dependent — and the dependency, solved exactly, is the polynomial. The same count shows every sum, product and quotient of algebraic numbers is algebraic, without ever needing a formula.

algebra · Field extensions

Named alongside it

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

Dimension Algebraic number Basis Conjugate Convex hull Convexity Counterexample Degree of an extension Exhaustive search Field extension Finite intersection Minimal polynomial

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