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

Column Space: Every Output Ax Lies on One Plane

Chapter 2 · Mathematics for AI and Machine Learning · Advanced · 1:44

An interactive live film: Ax = x0 a0 + x1 a1 drawn tip to tail in 3D stays on the plane C(A); a target b off the plane has no solution, and the reader drags Ax to find the shortest, perpendicular residual at the projection Ax̂.

Narration

Multiplying a matrix by a vector combines the columns of the matrix. Here A has two columns in three dimensional space: the red arrow a zero, and the blue arrow a one.

A times x takes x zero copies of a zero and x one copies of a one, and adds them tip to tail. The green arrow is the result, A x.

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 2Preview1:49Rank One as a LandscapeChapter 2Preview2:29Four Views of Matrix MultiplicationChapter 2Preview1:37The Projection Matrix: Outer times Inverse InnerChapter 2Preview1:36Matrix Norm and Condition Number: the Largest StretchChapter 2