1. $\pmatrix{a_1&a_2&a_3\cr a_4&a_5&a_6\cr a_7&a_8&a_9}\pmatrix{0\cr 0\cr 0} = \pmatrix{0+0+0\cr 0+0+0\cr 0+0+0} = \pmatrix{0\cr 0\cr 0}$
A matrix transformation takes the origin to the origin.
2. $M(\lambda \mathbf{a} + \mu \mathbf{b}) = M(\lambda \mathbf{a}) + M(\mu \mathbf{b}) = \lambda (M\mathbf{a}) + \mu (M\mathbf{b}) $
(Where $M$ is any matrix, $\mathbf{a}, \mathbf{b}$ are fixed vectors, $\lambda , \mu$ are scalars. $\lambda \mathbf{a} + \mu \mathbf{b}$ defines a plane. If $\mathbf{b} = \mathbf{o}$, this expression defines a line.)
Planes through the origin are transformed into other planes or lines through the origin or a point at the origin by matrix transformations. Lines are transformed into lines through the origin or a point at the origin.
3. $M(\lambda \mathbf{a} + \mu \mathbf{b} + \mathbf{c}) = M(\lambda \mathbf{a}) + M(\mu \mathbf{b}) + M\mathbf{c} = \lambda (M\mathbf{a}) + \mu (M\mathbf{b}) + M\mathbf{c}$
(Where $M$ is any matrix, $\mathbf{a}, \mathbf{b} \mathbf{c}$ are fixed vectors, $\lambda , \mu $ are scalars)
Planes not through the origin are transformed into planes or lines or points, through or not through the origin.
4. Any line - think about matrices of the form $\pmatrix{a_1 & a_2 & a_3 \cr 0 & 0 & 0 \cr 0 & 0 & 0}$
5. Any plane - think about matrices of the form $\pmatrix{a_1 & a_2 & a_3 \cr a_4 & a_5 & a_6 \cr 0 & 0 & 0}$
6. The example given for part 4 transforms everything into a line!
7. The result from part 2 shows this to be impossible.
8. Any combinations of enlargements, rotations and reflections. These are the only shape-preserving transformations.
9. Matrix transformations take parallel lines to parallel lines (or points) - think about the result of part 3, and the equations of parallel lines.
10. No - the origin does not move under a matrix transformation