Concept

Subspace

A subset of a vector space that is itself a vector space - closed under addition and under scaling, and so containing the origin. The set of everything a matrix can produce is one, which is why a system with no solution is a question about whether a point lies in a subspace rather than about arithmetic.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

The syndrome of 1011010, and the bit it names. A parity-check matrix over a received word, with the resulting syndrome matched against the table of single-error syndromes.

Finding the error without reading the message

Three parity checks on a seven-bit word produce three bits. If they are all zero nothing is wrong; otherwise they are the number of the position that broke. The message is never consulted, because the answer does not depend on it.

computation · Error-correcting codes

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

The nearest point of the plane the columns span. A target vector in space, the plane spanned by two columns, the point of that plane nearest the target, and the residual joining them, which meets the plane at a right angle.

The nearest point of a flat thing

More equations than unknowns almost never have a solution. Asking instead for the point of a plane nearest to where the answer should have been turns an unanswerable question into a shadow, and the shadow is what a line of best fit is.

algebra · Inner product

The map moved to the other side of the product. Two panels over the unit curve of the product u₁v₁ + u₂v₂. The left applies A to u and measures it against v, giving 2.328; the right applies the adjoint to v and measures it against u, giving the same number.

Moving a map across a product

The transpose looks like a fact about a matrix: reflect its entries in the diagonal. It is a fact about the inner product. Measure lengths and angles differently and the map that slides to the other side of the product is a different matrix, and a matrix that was symmetric stops being so.

algebra · Inner product

Nine bases that cannot each have one true direction. Eighteen vectors (0, 0, 0, 1) (0, 0, 1, 0) (1, 1, 0, 0) (1, −1, 0, 0) (0, 1, 0, 0) (1, 0, 1, 0) (1, 0, −1, 0) (1, −1, 1, −1) (1, −1, −1, 1) (0, 0, 1, 1) (1, 1, 1, 1) (0, 1, 0, −1) (1, 0, 0, 1) (1, 0, 0, −1) (0, 1, −1, 0) (1, 1, −1, 1) (1, 1, 1, −1) (1, −1, −1, −1); vector i joins bases 1–2, 1–5, 1–3, 1–7, 2–5, 2–8, 2–4, 3–4, 3–6, 3–7, 4–6, 4–8, 5–9, 5–6, 6–9, 7–8, 7–9, 8–9; near-solution true: 2, 6, 10, 11, basis B9 empty.

No way to hand out true and false

Birkhoff and von Neumann read the propositions of quantum mechanics as subspaces, with "and" the intersection and "or" the span. Three lines in a plane already break the distributive law. Kochen and Specker proved something stronger: the propositions cannot be called true and false at all, if every complete measurement is to have exactly one true outcome. Cabello's eighteen directions in four dimensions show why in one line of arithmetic — nine bases, each direction in two of them, and nine is odd — and a search of all 262,144 assignments confirms it.

logic · Non classical logic

Named alongside it

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

Orthogonality Inner product Least squares Residual Approximation Basis Dot product Gram schmidt Linear independence Orthonormal basis Parity Positive definite

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