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Matrix Transformation - A plane mapping onto itself.

  1. Mar 29, 2012 #1
    Find which planes map onto themselves under the matrx M.

    M=
    1 2 0
    0 1 -1
    0 2 1

    (in enclosed brackets - apologies for the format.).

    Attempt:

    Consider a plane ax+by+cz=d [1].

    M^-1 :
    3/3 -2/3 -2/3
    0 1/3 1/3
    0 -2/3 1/3

    (in enclosed bracket).

    - use of the inverse so that x,y,z can then be directly subbed into [1].

    x= X-2/3(Y)-2/3(Z)
    y=(Y)/3+(Z)/3
    z=-2(Y)/3 + Z/3

    - subbing this into [1] and multiplying throughout by 3 ,gives any plane maps to:

    3a(X) + (Y)(-2a+b-2c) + (Z)(-2a+b+c) [2]

    Here I am unsure how to interpret the comparison of the co-efficients between [1] and [2] of x,y and z:

    3a=a
    b= -2a+b-2c => a =c
    c= -2a +b+c => 2a=b

    I am unsure of what to do next...
     
  2. jcsd
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