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 3 essays across 2 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

Named alongside it

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

Approximation Basis Dot product Gram schmidt Inner product Least squares Linear independence Orthogonality Orthonormal basis Projection Residual Cauchy schwarz

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