Lesson 2.4 · 9 of 25 lessons
Independence & dependence
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
Neither a nor b is a scalar multiple of the other, so they provide two directions
- 02
αa + βb = 0 only when α=β=0
The two vectors are linearly independent - 03
Now consider a vector that is a multiple of the other
- 04
q is 2p, so it repeats the same direction
It adds no new direction, making the pair linearly dependent
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 1
coordinates on
2, 1 #a@
-1, 2 #b@
// Neither {{a}} nor {{b}} is a scalar multiple of the other, so they provide two directions
focus a b
checkpoint Independence & dependence · 1
// `α{{a}} + β{{b}} = 0` only when `α=β=0`\nThe two vectors are **linearly independent**
checkpoint Independence & dependence · 2
focus -
clear
1, 1 #p@
// Now consider a vector that is a multiple of the other
2, 2 #q@
// {{q}} is `2{{p}}`, so it repeats the same direction\nIt adds no new direction, making the pair **linearly dependent**
focus p q03 · Unit 2
Lessons in this unit
Vector spaces & transformations — Grow from linear combinations to independence, bases, the fundamental subspaces, and transformations.