Turn the rulers and every coordinate changes, though nothing moved. The right rulers make the data simple.
–Same points. Different numbers.
What changing basis is
A basis is a choice of rulers: two arrows you agree to measure everything against. Turn the rulers and every coordinate changes — but nothing moved. The dots are exactly where they were; only the description is different.
What it buys you
The right basis makes data simple. Aligned with the cloud’s own axes, almost all the spread sits on one coordinate and the other is nearly noise — that is what principal components, and the singular value decomposition two demos on, find automatically. In a network, every layer is a change of basis chosen by training.
Evidence. Points: the Bible word vectors, seen in two arbitrary dimensions of the learned 32 (raw2d in foundations/data/vectors.json), which is why the cloud sits tilted. “Snap” rotates the basis to the eigenvectors of the 2×2 covariance of these points, computed on the page; the share of variance on each axis is computed live from the coordinates you see.
Nine seconds elsewhere
The foundations in learning order, then the series they lead to. Each lands in about nine seconds.