How to find reflection of $(a,b)$ along $y=x, y = -x$

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How to find reflection of $(a,b)$ along $y=x, y = -x$

I know for $y=x$ it is $(b,a)$ and for $y=-x$ it is $(-b,-a)$. But how to prove it?

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Well, you can just use Basic Linear Algebra for instance just write the matrix of the transformation(Reflection in this case) with a usual basis in both the cases, then it will be very clear to you that you are just supposed to operate that matrix and obtain the desired result.