33. Change of Basis
The same vector or transformation gets different coordinate representations depending on which basis you use. Change of basis converts between two such representations.
33.1. Change of basis matrix
Given two bases and of the same space, the change-of-basis matrix from to is
— each column is the -basis vector expressed in -coordinates.
It satisfies:
The reverse direction uses the inverse:
33.2. Standard basis as anchor
For a basis of , write the basis vectors as columns of a matrix:
Then (standard ↔ conversion via plain matrix–vector product).
Example
in , (standard coords).
→ .
33.3. Change of basis for a linear transformation
If a linear transformation has matrix in basis and matrix in basis , with change-of-basis matrix (mapping coords to coords):
This is similarity. Two matrices related by for some invertible are similar — they represent the same linear transformation in different bases.
Similar matrices share:
- Determinant:
- Trace:
- Eigenvalues (with the same multiplicities)
- Rank
33.4. Why it matters
- Diagonalization is exactly the search for a basis in which is diagonal — making computation trivial.
- Many problems become easier in an eigenbasis, orthonormal basis, or specially-chosen adapted basis.