Matpic
mathematics, in pictures
Concept

Inner product

The operation pairing two vectors or functions into a number, from which length and the angle between them are defined.

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

abshadowa · b = 10.00= 2.43 × 4.12

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 −ππ-11× sin 1xarea = 4.000coefficient 1.273−ππ-11× sin 2xarea = 0no such harmonic in it−ππ-11× sin 3xarea = 1.333coefficient 0.424−ππ-11× sin 4xarea = 0no such harmonic in iteach panel is the target multiplied by one harmonic, with the area above the axis in one colour and the area below in the otheragainst its own harmonic the two do not cancel; against any other they cancel exactly, which is what makes one coefficient extractable without disturbing the rest

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.

analysis · fourier series

Named alongside it

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

OrthogonalityProjectionConvergenceCosineDot productFourier analysisHarmonicsMatrixSpectrumWork

All concepts