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Math Help - Find the matrix of reflection through the line of equation ax + by = 0 in ℝ

  1. #1
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    Find the matrix of reflection through the line of equation ax + by = 0 in ℝ

    Note: I'm using ";" to show in my matrices and vectors to represent the end of a row, e.g. [2; 4] is a column vector with 2 at the top and 4 at the bottom.

    I know how to find the matrix of reflection through y = x in ℝ (for which a = -b) by using the standard bases e1 = [1; 0], e2 = [0; 1] and got the matrix T = = [0 1; 1 0]
    I tried exploring with y = -x (for which a = b) and got T = [0 -1; -1 0].

    But I'm not seeing any pattern to a reflection across such a general case ax + by = 0.
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  2. #2
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    Re: Find the matrix of reflection through the line of equation ax + by = 0 in ℝ

    "Reflection in a line" maps any vector parallel to the line into itself. It maps any vector perpendicular to the line into its negative.

    So: What is a vector in the direction of the line ax+ by? What is a vector in the direction of ax+ by= 0? What is a vector perpendicular to it?


    Another way: find the rotation matrix, A, that rotates ax+ by= 0 to y= 0. The matrix that refleclects about y= 0 is R= \begin{bmatrix}1  & 0 \\ 0 & -1\end{bmatrix}. The matrix that reflects in the line ax+ by= 0 is A^{-1}RA.
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  3. #3
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    Re: Find the matrix of reflection through the line of equation ax + by = 0 in ℝ

    Hi,
    Please do not post a question more than once. The attachments provide a complete answer to your question as suggested by the previous response:

    Find the matrix of reflection through the line of equation ax + by = 0 in ℝ²-mhfgeometry37a.png
    Find the matrix of reflection through the line of equation ax + by = 0 in ℝ²-mhfgeometry37b.png
    Thanks from twilightmage13 and MINOANMAN
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