The interactive film is in the book's Companion

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The Rank of a Product: Dimensions Can Only Be Lost

Chapter 4 · Mathematics for AI and Machine Learning · Beginner · 1:30

An interactive live film: a cloud filling space goes through B and then A; each matrix can flatten it, neither can unflatten it, so rank(AB) ≤ min(rank A, rank B), and it is smaller still when A kills a direction that B produces.

Narration

On the left, a cloud of points fills all of space. The matrix B moves every point into the middle panel, and then the matrix A moves it on into the right panel.

Here B has rank two, so the middle cloud lies flat on a plane. Everything B produces lives in that plane.

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

More from Chapter 4

Interactive1:51Householder Reflection: A Mirror That Clears a ColumnChapter 4Preview1:56Householder QR: One Mirror per ColumnChapter 4Preview1:24Numerical Rank: When Is a Pancake a Plane?Chapter 4Preview1:39Gram-Schmidt or Householder: Losing the Right AnglesChapter 4Preview1:41A Rank-One Change of the Identity: Identity, Projection, MirrorChapter 4