I have a question that asks how a $2\times2$ real matrix $A$ can "transform" a complex number if we view the complex number as a column vector.
For example:
Find the matrix which can transform any complex number $z$ to $i\bar{z}$
So I know $i$ corresponds to a $90$ degree rotation anticlockwise and $\bar{z}$ corresponds to a reflection across the $x$-axis.
My question is which transformation comes first - the $90$ degree anticlockwise rotation OR the reflection across the $x$-axis? From there I can find the single transformation matrix by multiplying the matrices in the "reverse" order that they can come in).
What is the general rule regarding order of transformations for reflections, rotations, dilations?
Thanks in advance!
If you want $i\overline{z}=i(x-iy)=y+ix$, then first reflect w. respect to the real axis, then rotate counter-clockwise $90^\circ.$
Reversing the order results in $\overline{iz}=\overline{ix-y}=-y+ix$.
Anyway, to compute $i\overline{z}$:
$$\pmatrix{0&1\\1&0}\pmatrix{x\\y}=\pmatrix{y\\x},$$ so take $A$ to be $$A=\pmatrix{0&1\\1&0}.$$
Breaking this up into a reflection $R$ followed by a rotation $P$, so that $$A=PR:$$
$$R=\pmatrix{1&0\\0&-1}$$ $$P=\pmatrix{\cos 90^\circ & - \sin 90^\circ\\ \sin 90^\circ &\cos 90^\circ}=\pmatrix{0&-1\\1&0}$$ $$A=PR=\pmatrix{0&-1\\1&0}\pmatrix{1&0\\0&-1}$$ Again: $$A=\pmatrix{0&1\\1&0}.$$