Change of Basis

Start here

Back in Chapter 2 we said coordinates are not sacred; they are just the amounts you scale your chosen basis vectors by. This chapter cashes that in. Two people can describe the very same arrow with different numbers because they use different basis vectors. Change of basis is the dictionary that translates between them.

Two people, one arrow, different numbers

Imagine you and a friend named Jen looking at the same arrow in space.

• You use the standard basis: î = [1, 0] and ĵ = [0, 1].
• Jen uses her own basis vectors, which (in your coordinates) are b1 = [2, 1] and b2 = [-1, 1].

The arrow does not change. But because Jen measures with different rulers, she writes down different coordinates than you do. Neither of you is wrong. You are just speaking different coordinate languages.

The question of this chapter is simple: given Jen's numbers, how do we find your numbers, and the other way around?

The change of basis matrix

Here is the trick. Build a matrix whose columns are Jen's basis vectors, written in your coordinates:

B = | 2  -1 |     column 1 = b1 = Jen's first basis vector, in your coords
    | 1   1 |     column 2 = b2 = Jen's second basis vector, in your coords

Now think about what this matrix does using Chapter 3. Its columns are where the basis vectors land. So if you feed it a vector written in Jen's language, it returns that same arrow written in your language.

your coordinates = B * (Jen's coordinates)

It works because a vector in Jen's coordinates, say [x, y], literally means "x of b1 plus y of b2", and the matrix B computes exactly that combination in your coordinate system.

To go the other way, from your coordinates to Jen's, just use the inverse:

Jen's coordinates = B inverse * (your coordinates)

B translates Jen to you; B inverse translates you to Jen.

Translating a whole transformation

Now the powerful part. Suppose you have a transformation, like a 90 degree rotation, written as a matrix M in your coordinates. Jen wants the matrix for that same rotation in her coordinates. How?

You cannot just hand her M, because M expects and produces vectors in your language, not hers. So you build a translation sandwich. Read it right to left:

M in Jen's language = B inverse * M * B

Walk through what happens to a vector written in Jen's coordinates:

1. B translates it from Jen's language into your language.
2. M applies the transformation, in your language (where you know how to do it).
3. B inverse translates the result back into Jen's language.

The middle step is the actual work; the outer two just translate in and out. The whole expression is one matrix that performs the transformation entirely in Jen's coordinate system.

Why anyone would bother

Switching coordinate systems is not just for friends with quirky tastes. Sometimes a transformation is messy in the standard basis but becomes wonderfully simple in a smarter one. A rotation around a tilted axis is ugly in normal coordinates but trivial if you pick a basis lined up with that axis.

This is the entire motivation for the next chapter. If you can find a basis made of a transformation's eigenvectors, the transformation turns into a simple diagonal matrix, and hard calculations become easy.

Why this matters later

• Eigenbasis (Chapter 14) is a change of basis chosen to make a matrix as simple as possible.
• Principal Component Analysis in data science is literally a change of basis to the directions where data varies most.
• Physics and graphics constantly switch between world coordinates, camera coordinates, and object coordinates, each a change of basis.

Quick gotchas

The columns of B are the new basis vectors written in the old coordinates. Get this direction right or everything flips.

B goes from the new language to the old one. It feels backwards, but remember: it takes new coordinates in and gives old coordinates out.

Read the sandwich right to left. B first, then M, then B inverse.

What you walked away with

• The same arrow has different coordinates in different bases.
• The change of basis matrix B (new basis vectors in old coords) converts new coordinates to old; B inverse converts old to new.
• To rewrite a transformation in a new basis, use the sandwich B inverse * M * B.
• Choosing a clever basis can make a hard transformation simple.

Next up, Chapter 14: eigenvectors and eigenvalues, the special arrows a transformation only stretches without knocking off their line, and the reason a well-chosen basis can turn a matrix into a simple diagonal.