Lesson 2.7 · 12 of 25 lessons
Scaling & rotation
Grow from linear combinations to independence, bases, the fundamental subspaces, and transformations.
Open the interactive animation01 · Story
What to verify in this scene
Follow the explanations and operations, then identify the mathematical evidence that remains in the final scene.
- 01
Diagonal matrix S keeps the axis directions but doubles horizontal length
- 02
Apply S to widen the x-spacing of the basis and grid
- 03
Only the x direction stretched, so the space became wider
- 04
Now use rotation matrix R to turn the basis and grid by 90 degrees
- 05
Both axes and the grid rotated together
Matrix multiplication transforms the entire coordinate grid
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.
2d
zoom 4/5
grid on
relative-grid on
basis on
coordinates off
2 0 #S
0 1
// Diagonal matrix {{S}} keeps the axis directions but doubles horizontal length
checkpoint Scaling & rotation · 1
// Apply {{S}} to widen the x-spacing of the basis and grid
S
// Only the x direction stretched, so the space became wider
checkpoint Scaling & rotation · 2
clear
space reset
0 -1 #R
1 0
// Now use rotation matrix {{R}} to turn the basis and grid by 90 degrees
checkpoint Scaling & rotation · 3
R
// Both axes and the grid rotated together\nMatrix multiplication transforms the **entire coordinate grid**03 · Unit 2
Lessons in this unit
Vector spaces & transformations — Grow from linear combinations to independence, bases, the fundamental subspaces, and transformations.