Range & Rank
Range = every place a matrix can send things — the complete set of possible outputs. In the shadow picture, the range is the wall the shadow lands on: all the shapes a shadow could possibly be.
Rank = the dimensionality of the range — how many dimensions the outputs actually fill.
• Outputs fill the whole 2-D plane → rank 2.
• Outputs only ever land on a 1-D line → rank 1.
• Everything maps to a single point → rank 0.
Rank counts how many independent directions survive the transformation.
Watch the OUTPUT panel below. Drag the blue input dot r around and keep your eye on the green output g.
• With P, g roams the whole plane — the range is all of 2-D → rank 2.
• Switch to Q and drag: no matter where you drop r, g is stuck on a single green line — the range is 1-D → rank 1.
The red dashed line is the null space: every input on it lands on the same output. Drag r along it — the output g doesn't move! The output only ever lands on the green range line (rank 1). So given g, you can't tell which r it came from → not invertible, information lost.
🤖 Rank is how much real information gets through. Q's rank-1 output means its two LGN neurons together carry only one independent number — the second is redundant. This is the exact idea behind PCA and low-rank compression: the true signal often lives in far fewer dimensions than the raw data's size suggests.