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