Some matrices do something drastic: they squish space onto a lower dimension — flattening the whole 2-D plane onto a single line. It's like how a 3-D object casts a flat 2-D shadow on a wall. And once that shadow is cast, information is lost. This one idea unlocks everything in the next few lessons — so let's build it carefully.

Hold your hand up in front of a lamp. 🔦 Its shadow on the wall is a flat, 2-D outline of your 3-D hand. The shadow is a transformation of the hand — and it threw a whole dimension away. From the shadow alone, you can't rebuild the exact hand: many different hand poses cast the same shadow. A squishing matrix does exactly this to space. Keep this picture — it's the key to the whole lesson set.

A concrete squish. The matrix [[1, 0], [0, 0]] projects every point (x, y) straight down onto (x, 0) — flattening the entire plane onto the x-axis line. The up-down dimension is simply gone. Its determinant is 0: the unit square gets crushed to zero area.

See it happen. In the transformer below, pick Q (singular) and watch the whole grid collapse onto a single line (det = 0). Then pick P (invertible) — it stretches and rotates space but keeps it a full, 2-D plane. That difference is the entire story.

The question this raises — the one a neuroscientist actually cares about: if a neuron's weights squish the input like this, can we still figure out what the input was? To answer it precisely we need three words, each best understood with the shadow in mind: range, rank, and null space. One at a time, next. 👉