Editing Cartesian coordinate system (section)

====General matrix form of the transformations====
All [[affine transformation]]s of the plane can be described in a uniform way by using matrices. For this purpose, the coordinates <math>(x,y)</math> of a point are commonly represented as the [[column matrix]] <math>\begin{pmatrix}x\\y\end{pmatrix}.</math> The result <math>(x', y')</math> of applying an affine transformation to a point <math>(x,y)</math> is given by the formula
<math display=block>\begin{pmatrix}x'\\y'\end{pmatrix} = A \begin{pmatrix}x\\y\end{pmatrix} + b,</math>
where
<math display=block>A = \begin{pmatrix} A_{1,1} & A_{1,2} \\ A_{2,1} & A_{2,2} \end{pmatrix}</math>
is a 2×2 [[Square matrix|matrix]] and <math>b=\begin{pmatrix}b_1\\b_2\end{pmatrix}</math> is a column matrix.<ref>{{harvnb|Brannan|Esplen|Gray|1998|p=49}}.</ref> That is,
<math display=block>
\begin{align}
x' &= x A_{1,1} + y A_{1,1} + b_{1} \\
y' &= x A_{2,1} + y A_{2, 2} + b_{2}.
\end{align}
</math>

Among the affine transformations, the [[Euclidean transformation]]s are characterized by the fact that the matrix <math>A</math> is [[orthogonal matrix|orthogonal]]; that is, its columns are [[orthogonal vectors]] of [[Euclidean norm]] one, or, explicitly,
<math display=block>A_{1,1} A_{1, 2} + A_{2,1} A_{2, 2} = 0</math>
and
<math display=block>A_{1, 1}^2 + A_{2,1}^2 = A_{1,2}^2 + A_{2, 2}^2 = 1.</math>

This is equivalent to saying that {{math|''A''}} times its [[transpose]] is the [[identity matrix]]. If these conditions do not hold, the formula describes a more general [[affine transformation]].

The transformation is a translation [[if and only if]] {{math|''A''}} is the [[identity matrix]]. The transformation is a rotation around some point if and only if {{math|''A''}} is a [[rotation matrix]], meaning that it is orthogonal and
<math display=block> A_{1, 1} A_{2, 2} - A_{2, 1} A_{1, 2} = 1 .</math>

A reflection or glide reflection is obtained when,
<math display=block> A_{1, 1} A_{2, 2} - A_{2, 1} A_{1, 2} = -1 .</math>

Assuming that translations are not used (that is, <math>b_1=b_2=0</math>) transformations can be [[function composition|composed]] by simply multiplying the associated transformation matrices. In the general case, it is useful to use the [[augmented matrix]] of the transformation; that is, to rewrite the transformation formula 
<math display=block>\begin{pmatrix}x'\\y'\\1\end{pmatrix} = A' \begin{pmatrix}x\\y\\1\end{pmatrix},</math>
where 
<math display=block>A' = \begin{pmatrix} A_{1,1} & A_{1,2}&b_1 \\ A_{2,1} & A_{2,2}&b_2\\0&0&1 \end{pmatrix}.</math>
With this trick, the composition of affine transformations is obtained by multiplying the augmented matrices.