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Math Help - Linear Transformation & Invariant subpsce Question

  1. #1
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    Linear Transformation & Invariant subpsce Question

    The matrix A given below represents a linear transformation L : R2 --> R2 in
    the standard basis. Argue that this linear transformation only has R2 and {0}
    as invariant subspaces.
    A = 0 1
    -1 0
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  2. #2
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    Quote Originally Posted by flaming View Post
    The matrix A given below represents a linear transformation L : R2 --> R2 in
    the standard basis. Argue that this linear transformation only has R2 and {0}
    as invariant subspaces.
    A = 0 1
    -1 0
    R2 is two dimensional so other than R2 and {0} subspaces are one dimensional.

    Assume that there is 1-dimensional subspace with basic vector V=(a,b) invariant for action of A. It means that
    A*V=cV (1)
    where c is some constant (in other words cV is some element of our subspace).
    From (1) we arrive at the two conditions:
    b=ca & -a=cb
    which for any c has only one solution: V=0.
    But that means that there doesn't exist 1-dimensional subspace inv. for A. This finishes the proof: the only subspace apart from R2 invaraint for A is {0}
    Last edited by katastrofa_nadfioletu; October 26th 2008 at 01:33 PM.
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  3. #3
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    Notice, by the way, that A takes (1, 0) to (0, -1) and (0, 1) to (1, 0): it rotates the plane clockwise by 90 degrees. The only subsets of the plane that stay fixed under a rotation are the center of the circle, (0,0) and the entire plane.
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