This is a practice question for my upcoming exam.
I am finding it difficult to understand the approach and solve questions like these, especially when it comes to structured questions (non-multiple choice). Could someone please guide me on how I can solve such questions? I do not know how the solution is $C$ in the above question. I would also be really grateful if you could elaborate with your own examples, along with this one.
Your help will be appreciated.
On
Perhaps this will be a helpful way of looking at transformations:
The answer to C is $R_{-\pi/4}P_xR_{\pi/4}$ which means (reading from right to left) "first rotate 45˚ clockwise. Then, project onto the x-axis. Then, rotate 45˚ back in the other direction".
How can we show this is a projection onto the line $y = -x$? One thing we can do is check that it does the right thing to 1: a vector parallel to this line and 2: a vector perpendicular to it.
Start with a vector parallel to the line. The rotation puts it on the $x$-axis, the projection leaves it alone, and the next rotation puts it back. So, as expected, nothing happens to this vector.
Start with a vector perpendicular to the line. The rotation puts it on the $y$-axis, the projection maps it to zero, and the rotation leaves it at zero. So, as expected, the vector gets mapped to zero.
Try to see what happens to these vectors with other transformations.
Your matrix solution $M$ is such that $M\left( \begin{array}{c} 1 \\ -1 \end{array} \right) = \left( \begin{array}{c} 1 \\ -1 \end{array} \right)$ and $M\left( \begin{array}{c} 1 \\ 1 \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \end{array} \right)$