Slide the second column onto the first and the plane folds onto a line. Different words land on the same point, and no undo exists.
rank 2Every input has its own output — nothing is lost.
Full rank
A matrix is full rank when its columns point in genuinely different directions: the plane stays a plane, distinct inputs stay distinct, and there is an inverse — an “undo” that puts every point back. The determinant is not zero.
Losing rank
Drag the slider and the second column swings onto the first. The plane folds onto a line; words that were far apart land on the same point (the pairs shown in red), and no undo exists because the information about where they came from is gone. Rank counts how many independent directions survive.
Evidence. The matrix is M(t) = [[1, −0.3 + 0.9t], [0.5, 0.9 − 0.6t]]: the first column is fixed and the second slides from an independent direction to exactly 0.6 × the first, so det M(t) = 1.05 (1 − t) reaches zero at t = 1. Rank, determinant, the merged pairs (images closer than 0.03 units) and the inverse are computed live on the page; the dots are the Bible word vectors of the earlier demos.
Nine seconds elsewhere
The foundations in learning order, then the series they lead to. Each lands in about nine seconds.