Change of Basis

A basis fixes a coordinate system, and a problem stated in one basis is often easier to solve in another. Diagonalization rewrites a matrix in an eigenbasis; principal axes rewrite a quadratic form in an orthogonal one. Switching bases changes the coordinate vector of every point in a uniform way, and a single invertible matrix carries out the conversion.¹

Two coordinate vectors for one point

Let $\mathcal{B}=\{b_1,b_2\}$ and $\mathcal{C}=\{c_1,c_2\}$ be bases for a space $V$. The same vector $x$ has a $\mathcal{B}$-coordinate vector and a $\mathcal{C}$-coordinate vector, and they generally differ: each grid measures $x$ against its own axes.

Suppose the two bases are related by

The change-of-coordinates matrix

The construction generalizes to any two bases of an $n$-dimensional space.

$P_{\mathcal{C}\leftarrow \mathcal{B}}$ is the change-of-coordinates matrix from $\mathcal{B}$ to $\mathcal{C}$: left-multiplication converts $\mathcal{B}$-coordinates into $\mathcal{C}$-coordinates. To recover the column rule, read $P_{\mathcal{C}\leftarrow \mathcal{B}}[x{]}_{\mathcal{B}}$ as a linear combination of the columns; the result is a $\mathcal{C}$-coordinate vector, so the columns must be $\mathcal{C}$-coordinate vectors too.

Its columns are the coordinate vectors of an independent set, so they are independent and $P_{\mathcal{C}\leftarrow \mathcal{B}}$ is invertible. The inverse runs the conversion backward:

Change of basis in ${\mathbb{R}}^n$

When $\mathcal{C}=\mathcal{E}$ is the standard basis, $[b_j{]}_{\mathcal{E}}=b_j$, so the transition matrix is just the basis-vector matrix from coordinate systems:

Converting between two nonstandard bases of ${\mathbb{R}}^n$ needs the columns $[b_j{]}_{\mathcal{C}}$, each of which solves a system $[c_1\cdots c_n]y=b_j$. Solving all $n$ systems at once is a single row reduction.

Reducing the left block to $I$ applies $[c_1\cdots c_n{]}^{-1}$ to every column, and applied to $b_j$ that produces $[b_j{]}_{\mathcal{C}}$ in the right block.

An equivalent formula uses the two standard-conversion matrices. Since $P_{\mathcal{B}}[x{]}_{\mathcal{B}}=x$ and $[x{]}_{\mathcal{C}}=P_{\mathcal{C}}^{-1}x$,

For matrices larger than $2\times 2$, the single row reduction above is faster than forming $P_{\mathcal{C}}^{-1}$ and multiplying.

Working in an abstract space

The same construction handles bases of ${\mathbb{P}}_n$ or any finite-dimensional space by working through coordinate vectors.

Reversing this — expressing a standard polynomial in the $\mathcal{B}$-basis — uses $(P_{\mathcal{C}\leftarrow \mathcal{B}}{)}^{-1}=P_{\mathcal{B}\leftarrow \mathcal{C}}$. Change of basis is the coordinate-level version of the similarity transformations that rewrite a linear map when its underlying basis changes.