5.5 Coordinates and Change of Basis

A street address only makes sense once you know the street plan; a GPS position only makes sense once you know which reference frame it is measured in. Vectors are the same. The numbers we write for a vector depend on the basis we use to describe it. The standard basis is convenient for writing things down, but it is often not the best basis for thinking: an engineer studying a tilted ramp prefers axes along and across the slope, an audio engineer prefers "sum and difference" of the two stereo channels, and an image-compression algorithm prefers a basis in which most coordinates are almost zero.

In Section 5.3 we saw that a basis gives each vector a unique list of weights. This section turns those weights into a tool. We define the coordinate vector $[\mathbf{x}]_{\mathcal B}$, show how a single matrix converts between standard and $\mathcal B$-coordinates, and then how to move between any two bases — the algebra behind rotated axes, colour spaces and data compression.

Coordinates Relative to a Basis

The weights are unique by Theorem 5.11, which is exactly why the definition makes sense. Notice that the order of the basis vectors matters: swapping $\mathbf{b}_1$ and $\mathbf{b}_2$ swaps $c_1$ and $c_2$. For the standard basis $\mathcal E=\{\mathbf{e}_1,\dots,\mathbf{e}_n\}$ of $\mathbb{R}^n$, $[\mathbf{x}]_{\mathcal E}=\mathbf{x}$.

The change-of-coordinates matrix $P_{\mathcal B}$

Why. $P_{\mathcal B}[\mathbf{x}]_{\mathcal B}=c_1\mathbf{b}_1+\dots+c_n\mathbf{b}_n=\mathbf{x}$ by the definition of matrix–vector multiplication. The columns of $P_{\mathcal B}$ form a basis of $\mathbb{R}^n$, so $P_{\mathcal B}$ is invertible by the Invertible Matrix Theorem (Theorem 5.20). In practice, to find $[\mathbf{x}]_{\mathcal B}$ we usually row reduce $[P_{\mathcal B}\,|\,\mathbf{x}]$ rather than compute $P_{\mathcal B}^{-1}$.

Coordinates in other vector spaces

The coordinate mapping is just as useful in spaces such as $P_n$ or $M_{m\times n}$: once a basis is fixed, every polynomial or matrix becomes a column of numbers, and every computation can be done in $\mathbb{R}^n$.

This is the theorem that justified our trick in Section 5.3 of replacing polynomials by their coefficient columns. (A one-to-one, onto, linear map is called an isomorphism — Chapter 6 studies such maps.)

Changing Between Two Bases

Suppose a vector is known in $\mathcal B$-coordinates and we want its coordinates relative to another basis $\mathcal C$. We could go via the standard basis, but there is a single matrix that does the whole job. Since the coordinate mapping is linear, $$[\mathbf{x}]_{\mathcal C}=[c_1\mathbf{b}_1+\dots+c_n\mathbf{b}_n]_{\mathcal C}=c_1[\mathbf{b}_1]_{\mathcal C}+\dots+c_n[\mathbf{b}_n]_{\mathcal C}=\begin{bmatrix}[\mathbf{b}_1]_{\mathcal C}&\cdots&[\mathbf{b}_n]_{\mathcal C}\end{bmatrix}[\mathbf{x}]_{\mathcal B}.$$

Read the arrow from right to left: $P_{\mathcal C\leftarrow\mathcal B}$ eats $\mathcal B$-coordinates and produces $\mathcal C$-coordinates. When $\mathcal C$ is the standard basis $\mathcal E$ of $\mathbb{R}^n$, $P_{\mathcal E\leftarrow\mathcal B}=P_{\mathcal B}$.

Applications: Rotated Axes and Better Bases for Data

If the coordinate axes are rotated anticlockwise through an angle $\theta$, the new axes point along the unit vectors $\mathbf{u}_1=(\cos\theta,\sin\theta)$ and $\mathbf{u}_2=(-\sin\theta,\cos\theta)$. With $\mathcal B=\{\mathbf{u}_1,\mathbf{u}_2\}$, the point with standard coordinates $(x,y)$ has new coordinates $(x',y')=[\mathbf{x}]_{\mathcal B}$. Here $P_{\mathcal B}$ is a rotation matrix, and its inverse is simply its transpose (Chapter 8 explains why), so $$\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}\begin{bmatrix}x'\\y'\end{bmatrix},\qquad\begin{bmatrix}x'\\y'\end{bmatrix}=\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}.$$

Rotations are not the only useful change of basis. Whenever the standard coordinates of some data are strongly related to each other, a different basis can separate "what is common" from "what is different".

Exercises