Edit made on December 22, 2014 by ColinWright at 12:54:14
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Linear transformations in Euclidean n-space can be represented by square matrices of the order EQN:n x EQN:n.
For example, the matrix representing a reflection in the line EQN:y=x $y=x$ in 2-dimensional space looks like this:
* EQN:\left[\begin{matrix}0&1\\1&0\end{matrix}\right] $\left[\begin{matrix}0&1\\1&0\end{matrix}\right]$
... and the matrix representing an enlargement in 3-dimensional space, scale-factor EQN:k , $k,$ centred at the origin looks like this:
* EQN:\left[\begin{matrix}k&0&0\\0&k&0\\0&0&k\end{matrix}\right] $\left[\begin{matrix}k&0&0\\0&k&0\\0&0&k\end{matrix}\right]$
Successive transformations can also be represented by a single composite matrix.
If 3 transformations, represented by the matrices EQN:T_1, EQN:T_2 $T_1,\,T_2,$ and EQN:T_3, $T_3$ are performed
in that order, then the composite matrix would be the product EQN:T_3T_2T_1 $T_3T_2T_1$
(note that this is not necessarily in general the same as EQN:T_1T_2T_3 $T_1T_2T_3$ as matrix multiplication
is non-commutative).