How can I derive these $3$ fundamental $2\times 2$ matrices:

Question

1) The matrix that represents counterclockwise rotation about the origin by angle $\theta$

2) The matrix that represents flips across the line $x=y$

3) The matrix that represents flips across the $x$-axis

Of course, all these are basic and can be easily found by a Google search.

But, I want to work out how to derive it, so that there's a much better chance that I remember what they are, in case I need to use them.

So, I could start?

I want all $3$ matrices with respect to the standard ordered basis, $\{(1,0),(0,1)\}$.

I guess the main difficulty is not knowing explicit formulas for the action on the basis -- if I had formulas, then the matrix computation algorithm is trivial.

So, I would like to figure out the formulas ... geometrically, I think.

Any suggestions are welcome.

Thanks,

Answer 1

The first column is $F\left[\begin{array}{c}1\\0\end{array}\right]$. The second column is $F\left[\begin{array}{c}0\\1\end{array}\right]$.
So just work out where $(1,0)^T$ and $(0,1)^T$ end up.