The interactive film is in the book's Companion

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The Matrix Inverse: Squashed Means Gone

Chapter 2 · Mathematics for AI and Machine Learning · Beginner · 1:29

An interactive live film: A sends the unit cube to a box and its inverse sends it back; flatten the box and the smallest singular value goes to zero, two far-apart points land together, and the inverse stretches by 1/σ_min until it no longer exists.

Narration

The grey cube holds every input with coordinates between zero and one. The matrix A sends it to the teal box, whose edges are the three columns of A.

The inverse matrix sends the box back onto the cube. Every output came from exactly one input, so the trip can be undone.

The rest of the narration is in the book's Companion.

More from Chapter 2

Interactive1:40Change of Basis: One Vector, Two Sets of CoordinatesChapter 2Preview1:53A Matrix Moves the CubeChapter 2Preview1:49Rank One as a LandscapeChapter 2Preview2:29Four Views of Matrix MultiplicationChapter 2Preview1:44Column Space: Every Output Ax Lies on One PlaneChapter 2Preview1:37The Projection Matrix: Outer times Inverse InnerChapter 2Preview1:36Matrix Norm and Condition Number: the Largest StretchChapter 2