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Concept

Dot product

The number obtained from two vectors by multiplying matching coordinates and adding. It is the signed length of the shadow one casts on the other times the length of that other, so it is zero exactly at right angles, and every notion of angle in a space without a picture is defined from it.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

The dot product as a shadow. Two vectors and the shadow the first casts on the second. The shadow is 2.425 long and b is 4.123, so the dot product is 10.000.

The dot product is a shadow

Multiply the matching coordinates and add them up. That rule explains nothing, and it hides the fact that the answer is a length — how far one arrow reaches along another, times how long that other one is.

algebra · Inner product 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. Gram–Schmidt on two planar vectors, in 3 panels: the pair as given, the shadow of the second on the first, and the perpendicular pair that is left when the shadow is removed.

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.

Inner productOrthogonalityProjectionApproximationBasisCauchy schwarzGram schmidtLeast squaresLinear independenceOrthonormal basisPositive definiteResidual

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