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I've been working through my notes and I've come across something I don't really understand, I'm hoping someone can help. The question and the answer to the question are below.

$$f(x,y)=(y,x) \quad \text{and} \quad g(x,y)=(-y,x)$$ The composite $g \circ f$ is given by $$\begin{align} (g \circ f)(x,y) &= g(f(x,y)) \\ &= g(y,x) \\ &= (-x,y). \end{align}$$

I'm struggling to work out how the y and x change position. If I draw out the translation I can understand how they change position. But if I didnt draw out the 2 translations and just combined the translations, how would I know that they change position. Thanks for any help Regards Mike

1 Answers 1

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This is a very simple problem. It is of utmost importance that you understand the solution. Of course you could interpret $g\circ f$ geometrically as a reflection in the line $x=y$, composed with a $90^\circ$ rotation; but it is much quicker to check what $f$ and $g$ do to the coordinates of points ${\bf z}=(x,y)$. The first map $$f:\quad (x,y)\mapsto (u,v):=(y,x)$$ just interchanges the two coordinates. The second map $$g:\quad (u,v)\mapsto (-v,u)$$ puts the negative of the second coordinate $v$ as first and copies the first coordinate $u$ as second entry of the image point $g(u,v)$. It follows that $g\circ f$ has the total effect $$g\circ f:\quad (x,y)\mapsto(y,x)\mapsto(-x,y)\ ,$$ as claimed.