1. Linear combinations are recipes
A linear combination has the form
The vectors are ingredients. The coefficients are recipe amounts. Change the coefficients below and watch the result.
2. Movement interpretation
Use two movement directions. First walk along c₁v₁, then along c₂v₂. The final arrow is the linear combination.
3. Span: everything the vectors can build
The span of two vectors is the set of all possible linear combinations. The canvas samples many coefficient pairs.
4. Can we build the target?
Given two building blocks in the plane, can we find coefficients that reach a target vector?
Choose a target
Column view
If A has columns v₁ and v₂, then Ac is a linear combination of the columns.
Use the vectors from Section 1 as columns.
5. Convex combinations: mixtures
A convex combination of two points has form (1−t)u + tv with 0 ≤ t ≤ 1.
6. Signals as combinations of waves
A sound-like signal can be created by combining simple waves.
7. Images as high-dimensional vectors
Each small image below is a grid of numbers. The blended image is a linear combination of image vectors.
The formula is α Image A + (1−α) Image B.
8. High-dimensional recipes
Generate random building blocks in a high-dimensional space. Even when the ambient dimension is large, using only k building blocks creates at most a k-dimensional span.
9. Reflection
Write a short paragraph explaining the difference between a linear combination and a span.