2. The hidden explanation of extended matrices

The extended matrix for an affine transformation in R$^ 2$ is a $3 \times 3$ matrix. At first extended matrices may seem to be just an algebraic trick, but there is actually a geometrical interpretation.

The basic idea is simple: A homogeneous linear transformation can't move the origin. However, if you ``embed'' R$^ 2$ as the $z=1$ plane in R$^ 3$ and consider the homogeneous linear transformations on R$^ 3$ that take the $z=1$ plane only to itself, these homogeneous linear transformations can duplicate the effect of affine transformations on the $z=1$ plane. The $3 \times 3$ matrix you need is precisely the extended matrix of the affine transformation. There is no problem about moving the origin, because the ``origin'' of the $z=1$ plane is not the origin of R$^ 3$. This three-dimensional interpretation is normally ``hidden'', in that you don't have to think about it when you use extended matrices.

Figure illustrates how the idea works for the translation $T($x$) =$   x$+ (2,1)$, which has extended matrix ${\left [\begin{array}{ccc} 1&0&0\\ 0&1&0\\ 2&1&1 \end{array}\right ] }$. The transformation on R$^ 3 \rightarrow$   R$^ 3$ takes the $z=1$ plane to itself and in that plane produces the desired translation.

Figure: The hidden explanation.

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