What a Linear Map Changes
Key concept: A weight matrix W is a geometric transformation — it stretches, rotates, and reshapes the space the data lives in.
Where Demos 2 and 2.5 ask “what is the number?”, this demo asks “what does the map do to the space?” A 2×2 matrix W is applied to a grid, a unit circle, and a two-class point cloud, and the input and output spaces are shown side by side.
The output panel keeps a faded “ghost” of the untouched grid and draws a hairline from each point’s original position to where it lands after W, so the displacement is something you can see rather than infer. A separate panel draws the singular axes σ₁u₁ and σ₂u₂ at their true lengths, and histograms show how the data’s spread along the dominant axis changes before and after.
What to look for: switch on the animated morph to watch the grid shear and the circle stretch into an ellipse continuously — the growing length of the displacement hairlines is the linear map, and it grows fastest along the direction W stretches most.
Interaction is easiest on a laptop/desktop screen. On a small screen, you can also open the demo in its own tab at full size.