Transformations for 10

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?