Experiments / live

The shadow of a vector playground

A projection is a shadow made exact. The steel line through the origin is a fixed subspace — the column space C(A) of a line. Drag the ink vector b anywhere you like and watch its shadow b̂ = Pb slide along the subspace to stay the closest point to b, while the gold dashed residual b − b̂ stretches and shrinks but stubbornly keeps its right angle. Land b on the subspace and the residual vanishes. The full story — inner products, the projection matrix P = A(AᵀA)⁻¹Aᵀ, Gram–Schmidt and QR — is in the essay.

drag the ink vector b — its shadow follows

b̂ = (0.00, 0.00) ‖b − b̂‖ = 0.00

This playground is interactive and needs JavaScript. Here's the idea as a static picture: the vector b casts a shadow onto the subspace C(A); the projection b̂ is the closest point in it to b, and the dashed residual b − b̂ drops straight down, perpendicular to the whole subspace.