Lesson 2.3 · 8 of 25 lessons

Linear combinations

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 linear combination multiplies each vector by a coefficient and then adds them
02
First add one copy each of u and v
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Now change only the coefficient of u to 2
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The sum out moves with it
Changing coefficients this way creates a linear combination

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
coordinates on

1, 2  #u@
2, -1  #v@
// A **linear combination** multiplies each vector by a coefficient and then adds them
focus u v
checkpoint Linear combinations · 1

focus -
// First add one copy each of {{u}} and {{v}}
sum(u, v)  #out@
checkpoint Linear combinations · 2

// Now change only the coefficient of {{u}} to 2
u * 2
// The sum {{out}} moves with it\nChanging coefficients this way creates a **linear combination**

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.