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Math Help - orthonomal basis

  1. #1
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    orthonomal basis

    Let T be a linear transformation from R^2 to R^2. and T is represented by the matrix B=
    [(1/5)^(1/2) -2(1/5)^(1/2)]
    [2(1/5)^(1/2) (1/5)^(1/2)]
    with respect to the stantard basis of R^2.

    a) Is T isometry?
    b) does R^2 have an orthonormal basis for eigenvectors of T?

    i can do a) but need help on b). i dont really understand what it is asking. is it asking to find an orthonormal basis for eigen space of T?
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  2. #2
    Super Member PaulRS's Avatar
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    Quote Originally Posted by Kat-M View Post
    Let T be a linear transformation from R^2 to R^2. and T is represented by the matrix B=
    [(1/5)^(1/2) -2(1/5)^(1/2)]
    [2(1/5)^(1/2) (1/5)^(1/2)]
    with respect to the stantard basis of R^2.

    a) Is T isometry?
    b) does R^2 have an orthonormal basis for eigenvectors of T?

    i can do a) but need help on b). i dont really understand what it is asking. is it asking to find an orthonormal basis for eigen space of T?
    It asks whether there's an orthonormal base for \mathbb{R}^2 made out of eigenvectors of T

    What they want you to see is that, if you were working with complex numbers instead of real numbers, b) would hold -by the spectral theorem for Unitary Transformations-, but here it may not ( Try ! ).
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