Change of Coordinates

Introduction

Abstract   Given two ordered bases \(\mathcal{A}\) and \(\mathcal{B}\) for a subspace \(W\) of \(\RNrSpc{n}\), we show that there is a certain matrix which transforms coordinate vectors with respect to \(\mathcal{A}\) into coordinate vectors with respect to \(\mathcal{B}\).

Outline   Let's begin with an example: We know that ‘kilometer’ and ‘mile’ are units of measurement of length. On a line L with a point designated as the origin, we may therefore use either of these two units to obtain a basis for \(L\). Converting then between kilometers and miles amounts to converting coordinate vectors with respect to the km-basis into coordinate vectors with respect to the miles basis; and we know that this conversion is handled by a single factor:

(number of km's) = 1.6 x (number of miles)

In essence this explains what happens when we want to convert between coordinate vectors with respect to two different bases in a subspace \(W\) of \(\RNrSpc{n}\). Say, we have two such bases

\[\VSpcBss{B}=(\Vect{b}_1,\dots ,\Vect{b}_r) \quad\text{and}\quad \VSpcBss{C}=(\Vect{c}_1,\dots ,\Vect{c}_r)\]

Then a vector \(\Vect{x}\) in \(W\) can be expressed in exactly one way as a linear combination of \(\VSpcBss{B}\) and of \(\VSpcBss{C}\):

\[ \begin{array}{rcllrcl} \Vect{x} & = & s_1 \Vect{b}_1 + \cdots + s_r\Vect{b}_r &\quad \text{so} & \CoordVect{x}{B} = (s_1,\dots ,s_r) \\ \Vect{x} & = & t_1 \Vect{c}_1 + \cdots + t_r\Vect{c}_r &\quad \text{so} & \CoordVect{x}{C} = (t_1,\dots ,t_r) \\ \end{array} \]

To convert from \(\OrdVSpcBss{B}\)-coordinates to \(\OrdVSpcBss{A}\)-coordinates, use the coordinate vectors of \(\OrdVSpcBss{B}\) with respect to \(\OrdVSpcBss{C}\) to construct the matrix

\[ \CoordTrafoMtrx{C}{C}{B} \DefEq \left[\begin{array}{cccc} \uparrow & \uparrow & & \uparrow \\ (\Vect{b}_1)_{\OrdVSpcBss{C}} & (\Vect{b}_2)_{\OrdVSpcBss{C}} & \cdots & (\Vect{b}_r)_{\OrdVSpcBss{C}} \\ \downarrow & \downarrow & & \downarrow \end{array}\right] \]

We then have the coordinate conversion identity

\(\Vect{x}_{\OrdVSpcBss{C}}\)\(=\)\(\CoordTrafoMtrx{C}{C}{B} \Vect{x}_{\OrdVSpcBss{B}}\)

Such coordinate conversion matrix is always invertible, and it satisfies

\[\CoordTrafoMtrx{C}{B}{C} = \left( \CoordTrafoMtrx{C}{C}{B} \right)^{-1}\]

This follows from the transitivity property of coordinate vector conversion:

\(\CoordTrafoMtrx{C}{D}{B}\)\(=\)\(\CoordTrafoMtrx{C}{D}{C} \CoordTrafoMtrx{C}{C}{B}\)

Here we are given three bases \(\OrdVSpcBss{B},\OrdVSpcBss{C},\OrdVSpcBss{D}\) of \(W\), and the above identity asserts that the conversion from \(\OrdVSpcBss{B}\)-coordinates to \(\OrdVSpcBss{D}\)-coordinates may just as well be accomplished by first converting from \(\OrdVSpcBss{B}\)-coordinates to \(\OrdVSpcBss{C}\)-coordinates and then further to \(\OrdVSpcBss{D}\)-coordinates.

PropositionChange coordinates matrix

Given ordered bases \(\OrdVSpcBss{B}=(\Vect{b}_1,\dots ,\Vect{b}_r)\) and \(\OrdVSpcBss{C}=(\Vect{c}_1,\dots ,\Vect{c}_r)\) of a vector space \(\VSpc{W}\), let

\[ \CoordTrafoMtrx{C}{C}{B}\ =\ \left[\begin{array}{cccc} \uparrow & \uparrow & & \uparrow \\ (\Vect{b}_1)_{\OrdVSpcBss{C}} & (\Vect{b}_2)_{\OrdVSpcBss{C}} & \cdots & (\Vect{b}_r)_{\OrdVSpcBss{C}} \\ \downarrow & \downarrow & & \downarrow \end{array}\right] \]

Then, for every \(\Vect{x}\) in \(W\),

\(\Vect{x}_{\EuScript{C}}\)\(=\)\(\CoordTrafoMtrx{C}{C}{B} \Vect{x}_{\OrdVSpcBss{B}}\)

Moreover, the matrix \(\CoordTrafoMtrx{C}{C}{B}\) with this property is unique.

The matrix \(\CoordTrafoMtrx{C}{C}{B}\) is called the change of coordinates matrix or the coordinate conversion matrix from \(\OrdVSpcBss{B}\)-coordinates to \(\OrdVSpcBss{C}\)-coordinates.

Let us now turn to rules which apply to coordinate conversion matrices: Suppose we are given three ordered bases of a subspace \(W\) of \(\RNrSpc{n}\), say \(\OrdVSpcBss{B}\), \(\OrdVSpcBss{C}\), and \(\OrdVSpcBss{D}\). We then have associated coordinate conversion matrices

\(\CoordTrafoMtrx{C}{C}{B}\)

  to convert from \(\OrdVSpcBss{B}\)-coordinates to \(\OrdVSpcBss{C}\)-coordinates

\(\CoordTrafoMtrx{C}{D}{C}\)

  to convert from \(\OrdVSpcBss{C}\)-coordinates to \(\OrdVSpcBss{D}\)-coordinates

\(\CoordTrafoMtrx{C}{D}{B}\)

  to convert from \(\OrdVSpcBss{B}\)-coordinates to \(\OrdVSpcBss{D}\)-coordinates

The relationship between these matrices is given by the following:

CorollaryProperties of coordinate change

Given ordered bases \(\OrdVSpcBss{B}\), \(\OrdVSpcBss{C}\), and \(\OrdVSpcBss{D}\) of a subvector space \(\VSpc{W}\) of \(\RNrSpc{n}\), the associated coordinate conversion matrices are related by

\(\CoordTrafoMtrx{C}{D}{B}\)\(=\)\(\CoordTrafoMtrx{C}{D}{C} \CoordTrafoMtrx{C}{C}{B}\)
CorollaryReversing coordinate change

If \(\OrdVSpcBss{B}\) and \(\OrdVSpcBss{C}\) are ordered bases of a subvector space \(\VSpc{W}\) of \(\RNrSpc{n}\), then

\(\CoordTrafoMtrx{C}{B}{C}\)\(=\)\(\left(\CoordTrafoMtrx{C}{C}{B}\right)^{-1}\)

Study Materials