Lesson 2.6 · 11 of 25 lessons

Column space & null space

Grow from linear combinations to independence, bases, the fundamental subspaces, and transformations.

Open the interactive animation

01 · Story

What to verify in this scene

Follow the explanations and operations, then identify the mathematical evidence that remains in the final scene.

01
A removes the z direction from three-dimensional space
02
Collapse the volume and relative grid to see where outputs can reach
03
The volume became zero and every output lies in the xy-plane
This plane is the column space of A
04
Now feed the z-axis vector n into the same A
05
n disappeared completely into the origin
Input directions sent to zero form the null space

02 · Notebook

The runnable notebook

This source creates the mathematical objects and controls the order of operations and animation. Read it here, then play the same scene yourself.

3d
zoom 4/5
grid off
relative-grid on
coordinates off
basis on

1 0 0  #A
0 1 0
0 0 0

det(i, j, k)  #volume@
// {{A}} removes the z direction from three-dimensional space
checkpoint Column space & null space · 1

// Collapse the volume and relative grid to see where outputs can reach
A with view 2d
// The volume became zero and every output lies in the xy-plane\nThis plane is the **column space** of {{A}}
checkpoint Column space & null space · 2

clear
space reset
view 3d
basis off
coordinates on
0, 0, 2  #n@
// Now feed the z-axis vector {{n}} into the same {{A}}
checkpoint Column space & null space · 3

A
// {{n}} disappeared completely into the origin\nInput directions sent to zero form the **null space**
focus n

Open the interactive animation

03 · Unit 2

Lessons in this unit

Vector spaces & transformations — Grow from linear combinations to independence, bases, the fundamental subspaces, and transformations.