Edit made on December 22, 2014 by ColinWright at 12:54:14

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~~WW~~ WM

HEADERS_END

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).