Find matrix representation of transformationGiven two lines l 1 : y = x − 3 and l 2 : x = 1 find matrix representation of transformation f(in standard base) which switch lines each others and find all invariant lines of f.

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Find matrix representation of transformationGiven two lines       l    1    :  y  =  x  −  3 and       l    2    :  x  =  1 find matrix representation of transformation   f(in standard base) which switch lines each others and find all invariant lines of   f.

eurgylchnj · Accepted Answer

Since the two straight lines intersect at the point   P  =  (  1  ,  −  2  ) the transformation must have a fixed point in   P. We can find such transformations in three steps.1) translate the origin in   P with the translation       T    P          −      1        (  x  ,  y  )  →  (  x  −  1  ,  y  +  2  )2) perform a simmetry   S with axis the strignt line passing thorough the new origin ad such that bisect the angle between the two lines.3) return to the old origin with the translation       T    P    (  x  ,  y  )  →  (  x  +  1  ,  y  −  2  ).So the searched matrix has the form:   M  =      T    P    S      T    P          −      1      .This is not a linear transformation but an affine one, and, if you want, can be represented by a   3  ×  3 matrix.

icedagecs · Accepted Answer

I have an addition:The translation matrices in omogeneous coordinate are:      T    P    =      [                            1                          0                          −          1                                      0                          0                          2                                      0                          0                          1                      ]          T    P          −      1        =      [                            1                          0                          1                                      0                          0                          −          2                                      0                          0                          1                      ]  and the reflection matrix can be found noting that the angle between the bisetrix and the   x-axis is   θ  =                    3        π            8      , Then the matrix is:  S  =      [                            cos          ⁡          2          θ                          sin          ⁡          2          θ                          0                                      sin          ⁡          2          θ                          −          cos          ⁡          2          θ                          0                                      0                          0                          1                      ]  Note that the invariant lines are the bisector and his orthogonal in   P.

Find matrix representation of transformation Given two lines l 1 </msub> : y =

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