Question

Are there any easy and good mnemonic for remembering how to do matrix transformations?

Answer 1

I think the best way is to simply understand them, since they're really not so bad if you're comfortable with matrix multiplication.

What you can do is create your matrix by putting in unit "test vectors" to create the matrix you want. I'll show you:

So for this example you want to create a matrix that rotates clockwise by 90 degrees. So, we know that our transformation will take a vector like this over here to this: So our matrix equation looks like this: \[\LARGE A \left(\begin{matrix}1 \\ 0\end{matrix}\right)=\left(\begin{matrix}0 \\ -1\end{matrix}\right)\] Now on your own paper you should probably write something like a blank matrix:\[\LARGE \left[\begin{matrix} \ \ \ & \ \\ \ & \ \end{matrix}\right] \left(\begin{matrix}1 \\ 0\end{matrix}\right)=\left(\begin{matrix}0 \\ -1\end{matrix}\right)\] So what this is realy saying is we're taking one of the first column and adding it to zero of the second column, so that literally tells us the first column is just the thing we have on the right there. \[\LARGE \left[\begin{matrix} 0 \ & \ \ \ \\ -1 & \ \end{matrix}\right] \left(\begin{matrix}1 \\ 0\end{matrix}\right)=\left(\begin{matrix}0 \\ -1\end{matrix}\right)\] Now make another picture like we did earlier, but choose the other obvious choice for a test vector: So we plug it on in: \[\LARGE \left[\begin{matrix} 0 \ & \ \ \ \\ -1 & \ \end{matrix}\right] \left(\begin{matrix}0 \\1\end{matrix}\right)=\left(\begin{matrix}1 \\ 0\end{matrix}\right)\] And now we see that the other column is now the answer: \[\LARGE \left[\begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix}\right] \left(\begin{matrix}0 \\1\end{matrix}\right)=\left(\begin{matrix}1 \\ 0\end{matrix}\right)\] So there you have it. Really simple to do in practice since there are usually some very nice vectors like this that you will know how they transform without even thinking about it much. If you have doubts that this is the rotation matrix: \[\LARGE A=\LARGE \left[\begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix}\right] \] then you might consider \[\LARGE A^2=-I\]which corresponds to a 180 degree rotation. Or you could consider that \[\LARGE -A\] is the rotation in the opposite direction so of course we know that \[\LARGE -A*A=I\] because we're just rotating back and forth to where we started. We can do something similar for the other matrices but I think it's much better to just think of them intuitively rather than try to find a mnemonic to get by with. If you want me to give you hints or help with those I can. I understand that maybe this example seems a little simplistic but you really can use this method for all the other transformations.