Lesson 3.2 · 14 of 25 lessons
Projection
Carry the right angle created by a dot product into projection and least-squares error.
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
A projection is the shadow of a vector on an axis or subspace
- 02
P removed the y component and lowered v to (3, 0)
- 03
Apply P again to a result that already lies on the x-axis
- 04
Nothing changes the second time
That is why a projection matrix satisfies P² = P
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 9/10
grid off
relative-grid on
coordinates on
1 0 #P
0 0
3, 2 #v@
// A **projection** is the shadow of a vector on an axis or subspace
checkpoint Projection · 1
P
// {{P}} removed the y component and lowered {{v}} to `(3, 0)`
focus v
checkpoint Projection · 2
focus -
// Apply {{P}} again to a result that already lies on the x-axis
P
// Nothing changes the second time\nThat is why a projection matrix satisfies `P² = P`
focus v03 · Unit 3
Lessons in this unit
Orthogonality, projection & least squares — Carry the right angle created by a dot product into projection and least-squares error.