Generator

The dot product as a shadow

A generator in the algebra library, called 30 times across 7 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

projection is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

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.

`show: "gram"`

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.

`show: "qr"`

The table Gram–Schmidt fills in, and the half it leaves empty. A 3-by-3 table of the coefficients writing each original vector in the orthonormal basis built from it. Everything above the diagonal is zero, which is the triangular shape the construction forces.

`show: "orthloss"`

Two orders for one subtraction, and only one of them survives. The worst inner product between two vectors that Gram–Schmidt has just made perpendicular, against the conditioning of the input, for the classical and the modified orders of the same subtractions.

`show: "legendre"`

The powers of x, straightened. The first 4 polynomials Gram–Schmidt returns when it is given 1, x, x², … and the inner product is an integral over [−1, 1]. They are the Legendre polynomials, which nobody was aiming at.

`show: "signs"`

The sign of a dot product is the side the shadow falls on. One vector held fixed and another swung through five angles. The dot product is positive while the shadow points along b, zero at a right angle, and negative beyond it.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

basis vectors 1 and 2 are perpendicular ×12
the polynomials of degree 0 and 1 are exactly perpendicular ×10
the polynomial of degree 0 is monic ×5
basis vector 1 has length one ×4
the degree-0 result is the Legendre polynomial scaled to be monic ×4
vector 1 is rebuilt from its own row of the table ×4
moving coefficient 0 by -0.25 makes the total worse ×3
moving coefficient 0 by 0.25 makes the total worse ×3
the 1th vector uses no direction built after it ×3
the residual is perpendicular to the column of x^0 ×3
vector 1 is two finite coordinates no larger than twelve ×2
a non-zero vector has positive weighted length ×1
a rational is two whole numbers ×1
a squared length is never negative ×1
a·b is |a||b| cos θ ×1
a·b is the shadow's length times |b| ×1
and is not perpendicular in the ordinary sense unless the weights agree ×1
and it is linear in its first slot ×1
and so has the second ×1
and to the second ×1
between one and three stages ×1
between three and forty points ×1
between two and five columns ×1
between two and four vectors go in ×1
both weights are positive and no larger than twenty-five ×1
degree between one and five ×1
each column is three finite coordinates ×1
every neighbouring point of the plane is further from the target ×1
every point is a finite pair ×1
every point of the drawn curve has weighted length one ×1
every vector has the same number of finite coordinates ×1
every ε is between nought and one ×1
no division by nought ×1
no sampled value falls below the vertex ×1
on the smallest ε drawn, the classical order loses orthogonality outright ×1
removing a shadow never lengthens a vector ×1
the camera angles are within a half turn ×1
the classical order is never the better of the two ×1
the degree is 1, 2 or 3 ×1
the dropped line meets b at a right angle ×1
the first basis vector has been scaled to length one ×1
the first vector is not zero ×1
the inequality holds for the weighted product too ×1
the least value is zero exactly when the two vectors are parallel ×1
the mode of the projection family is one of shadow, signs, schwarz, gram, qr, least, columns, weight, orthloss, legendre ×1
the modified order stays near the arithmetic's own precision ×1
the normal equations have a solution ×1
the nudged fit has one coefficient per column ×1
the nudged line is worse than the fitted one ×1
the pair still spans the same area, so it spans the same plane ×1
the parabola's least value is |a|² − (a·b)²/|b|² ×1
the plotted span is between 0.2 and 6 ×1
the quadratic cannot have two distinct real roots ×1
the residual is perpendicular to the first column ×1
the second drawn vector is perpendicular in the weighted sense ×1
the sign of a·b follows the angle ×1
the target is three finite coordinates ×1
the target's squared length splits into the projection's and the residual's ×1
the two columns are independent, or they span a line rather than a plane ×1
the two vectors are not parallel, or there is nothing left after the subtraction ×1
the vector being bounded is two finite coordinates no larger than twelve ×1
the vector it is measured against is two finite coordinates no larger than twelve ×1
the vector projected onto is not zero ×1
the vectors are independent, so nothing vanishes at its own step ×1
the weighted product does not care about the order ×1
there are at least as many coordinates as vectors ×1
there are more points than coefficients, or nothing is being fitted ×1
two columns span the plane ×1
two vectors go in ×1
what is left after the subtraction is perpendicular to the first vector ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Algebra

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

Symmetry forces a right angle

A matrix equal to its own reflection across the diagonal always has real stretches and always has perpendicular directions to stretch along. Neither is true of matrices in general, and both follow from one line of algebra.

Algebra

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

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

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

What a map throws away

A linear map redraws the grid, and the determinant measures how much it stretches area. When that measurement comes out zero the map has flattened the plane onto a line — and the question worth asking is not how much was lost but how much survived, because the two always add to what there was.

Analysis

Where the coefficients come from

The recipe for a square wave has a four over pi in front and a one over three on the second term, and the first rung of this ladder used them without saying where they came from. They come from multiplying by one harmonic and taking the area.

The whole library · What the figures prove