The interactive film is in the book's Companion

Play it, pause it, turn it and drag it with Mathematics for AI and Machine Learning.

Get the book

Rank One as a Landscape

Chapter 2 · Mathematics for AI and Machine Learning · Advanced · 1:49

An interactive live film: a 4 x 5 matrix as 3D bars; the outer product u v^T gives every row the same profile (rank one), a second outer product breaks it (rank two); find and drag the one wrong bar until the rank is one again.

Narration

Here is a matrix W with four rows and five columns, drawn as a landscape: every entry is a bar, and its height is the number.

This W is an outer product, u times v transpose. The vector u stands along the left edge, v along the back edge, and the bar in row i and column j is u i times v j.

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 2Preview1:29The Matrix Inverse: Squashed Means GoneChapter 2Preview1:53A Matrix Moves the CubeChapter 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