Concept

Orthonormal basis

A basis whose vectors are mutually perpendicular and each of length one. Its value is that a coordinate becomes a single inner product rather than the solution of a system, so coefficients can be computed independently of each other and a truncated expansion is the best approximation available from the directions kept.

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

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 Projection Residual Subspace Cauchy schwarz

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