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# Transformations for 10

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Age 16 to 18

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You can start by thinking about what happens to the origin under a matrix transformation by considering $\pmatrix{a_1&a_2&a_3\cr a_4&a_5&a_6\cr a_7&a_8&a_9}\pmatrix{0\cr 0\cr 0}$.

Where must the origin map to under a matrix transformation?

Can more than one point map to the origin under a matrix transformation? For example, you could consider the multiplication $\pmatrix{1 & 0 & -1\cr 0 & 1 & 1\cr 1 & 1 & 0}\pmatrix{x\cr y\cr z}$ and find some values of $x, y$ and $z$ which are mapped to the origin.

The equations for a line and plane in vector form may be useful.

Line: ${\bf r}={\bf a} + \lambda{\bf b}$

Plane: ${\bf r}={\bf a} + \lambda{\bf b}+ \mu{\bf c}$

Note that if ${\bf a} = {\bf 0}$ then the line or plane passes through the origin.

It may also be useful to recall that matrix multiplication is distributive:

${\bf M}({\bf a} + {\bf b}) = {\bf Ma} + {\bf Mb}$, and also that we have ${\bf M}(\lambda {\bf a}) = \lambda {\bf M}{\bf a}$.

You might like to consider what happens to parallel lines by considering ${\bf M}(\lambda {\bf b} + {\bf a_1})$ and ${\bf M}(\mu {\bf b} + {\bf a_2})$. What does this imply must happen to a square?

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

Follow hints using a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.

Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.