Hi I have a question which I don't really understand and wondered whether I could get some help on it.
Question:
Write the standard matrices representing the following linear transformations in the plane $R^2$:
$T_1:$ a reflection through the x-axis,
$T_2:$ an anti-clockwise rotation by $\theta$.
Hence find a value of $\theta$ satisfying $0 < $ $\theta$ $<$ $π$, for which $T_1$ $\circ$ $T_2$ $=$ $T_2$ $\circ$ $T_1$?
Any help would be grateful. Thanks.
I answer you one part of your question, you try the others...perhaps going over your notes first:
$$T_1\binom xy=\binom x{-y}\implies [T_1]=\begin{pmatrix}1&0\\0&\!-1\end{pmatrix}$$
Now, you should make an effort to understand (1) why the above definition of $\;T_1\;$ is the correct one, and (2) why the above matrix represents $\;T_1\;$ wrt the standard basis.
Once you've done the above, try the rest of your question. If you get stuck write back...