I was given the following question:
Find the $3 \times 3$ matrix that produces the described composite $2$D transformation below, using homogeneous coordinates. Translate by $(-3,2)$, and then scale the $X$-coordinate by $0.2$ and the $Y-$coordinate by $1.2.$
I have the answer key and don't need to send in an answer, so I'm not simply looking for the solution. I want to know how to solve it so I can solve similar problems.
Can anyone help me out at all?
I would like to be able to solve this kind of problem, but I don't understand the method of solving it.
A translation by $(a,b)$ should transform the point $(x,y,1)$ in homogenous coordinates to the point $(x+a,y+b,1)$. This is the linear transformation described by the matrix
$$\begin{pmatrix}1&0&a\\0&1&b\\0&0&1\end{pmatrix}.$$
So that's the matrix you need to apply to the homogenous coordinates to get a translation by $(a,b)$. And then scaling is the same as in the affine space, just keep the third coordinate constant. So scaling the $x$-coordinate by $c$ and the $y$-coordinate by $d$ is described by the matrix $\operatorname{diag}(c,d,1)$. The rest is just the usual matrix multiplication.