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 $𝐵=\left\{{\overrightarrow{𝑏}}_1,\dots ,{\overrightarrow{𝑏}}_{𝑛}\right\}$ and ${𝐵}^{\prime }=\left\{{\overrightarrow{𝑏}}_1^{\prime },\dots ,{\overrightarrow{𝑏}}_{𝑛}^{\prime }\right\}$ of the same space, the change-of-basis matrix from ${𝐵}^{\prime }$ to $𝐵$ is

— each column is the ${𝐵}^{\prime }$-basis vector expressed in $𝐵$-coordinates.

It satisfies:

The reverse direction uses the inverse:

33.2. Standard basis as anchor

For a basis $𝐵=\left\{{\overrightarrow{𝑏}}_1,\dots ,{\overrightarrow{𝑏}}_{𝑛}\right\}$ of ${ℝ}^{𝑛}$, write the basis vectors as columns of a matrix:

Then ${\left[\overrightarrow{𝑥}\right]}_{𝐸}=𝑃{\left[\overrightarrow{𝑥}\right]}_{𝐵}$ (standard ↔ $𝐵$ conversion via plain matrix–vector product).

33.3. Change of basis for a linear transformation

If a linear transformation has matrix $𝐴$ in basis $𝐵$ and matrix ${𝐴}^{\prime }$ in basis ${𝐵}^{\prime }$, with change-of-basis matrix $𝑃$ (mapping ${𝐵}^{\prime }$ coords to $𝐵$ coords):

This is similarity. Two matrices related by ${𝐴}^{\prime }={𝑃}^{-1}𝐴𝑃$ for some invertible $𝑃$ are similar — they represent the same linear transformation in different bases.

Similar matrices share:

• Determinant: $det\left({𝐴}^{\prime }\right)=det\left(𝐴\right)$
• Trace: $\text{tr}\left({𝐴}^{\prime }\right)=\text{tr}\left(𝐴\right)$
• Eigenvalues (with the same multiplicities)
• Rank

33.4. Why it matters

• Diagonalization is exactly the search for a basis $𝑃$ in which ${𝐴}^{\prime }={𝑃}^{-1}𝐴𝑃$ is diagonal — making computation trivial.
• Many problems become easier in an eigenbasis, orthonormal basis, or specially-chosen adapted basis.

33.5. See also

• Coordinate Vector
• Basis
• Diagonalization
• Eigenvectors & Eigenvalues