Lesson 31 of 48
Basis and change of basis
The same vector, described from a different point of view.
Coordinates are a choice, not a property. Every technique that “finds structure” — PCA, SVD, embeddings — is choosing a better basis and then saying the same thing more cheaply.
Do this
Write one vector's coordinates in two different bases by hand. Build the change-of-basis matrix and its inverse, and show that converting there and back returns the original.
The question that unlocks the next lesson
You change basis. What happens to the underlying vector?
- AIt rotates
- BNothing — only its coordinates change
- CIt is scaled by the determinant
- DIt becomes a different vector entirely