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I am working through a linear algebra text and I am doing exercises in a chapter dealing with linear transformations and linear functions. I am trying to come up with the transformation matrix for a vector reflected on a line in two-dimensional space. Please try to the answer the question without change of basis or eigenvalues as I have not reached that chapter. I am confident this does not warrant complicated trigonometry either.

So far I have projected the basis vectors of the space onto the line and found an orthogonal line in search of a solution. I am not sure where to go from here. Thank you

qq4
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    Have you tried drawing a diagram containing any arbitrary vector and line of reflection? Here is a rough sketch. – Andrew Chin Nov 14 '19 at 21:54
  • @AndrewChin yes, I just came to the geometric conclusion from a similar drawing. Thank you for confirming my findings. Now I am constructing a projection matrix by first finding a basis for the line and projecting the basis vectors on it. Then I should be able to find the reflection matrix by following the equation for a reflection. Is this correct? – qq4 Nov 14 '19 at 22:09
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    Sounds like the right idea. – Andrew Chin Nov 14 '19 at 22:23

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Consider a rohumbus $ABCD$ with the ceneter point $O$

Let the diagonal $AC$ being the line and the vector $AB$ being reflected to the vetor $AD.$

The projection is vector $AO$ which is half of vectro $AC.$

The reflection vector $AD$ is equal to the vector $BC$ and we have $$BC=AC-AB =2AO -AB$$

Mohammad Riazi-Kermani
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