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    Eigenvectors, eigenvalues of a transformation

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    Suppose a linear transformation is defined by [see the attachement for the equation] where x is a vector in the euclidean plane (R^2). Answer the following:
    a) What are the eigenvalues and eigenvectors of this transformation?
    b) Show that the image of x can be represented as a sum of the eigenvectors.
    c) What does this transformation represent?

    © BrainMass Inc. brainmass.com December 24, 2021, 10:22 pm ad1c9bdddf
    https://brainmass.com/math/linear-algebra/eigenvectors-eigenvalues-transformation-469484

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    SOLUTION This solution is FREE courtesy of BrainMass!

    From the condition, we have , where .
    (a) We set
    So the transform has two eigenvalues 1 and -1.
    When the eigenvalue is , the eigenvector is
    When the eigenvalue is , the eigenvector is
    (b) Consider , then we have

    Then we get ,
    So the image of is a linear combination of two eigenvectors.
    (c) This transform maps to , maps to . Let , then the transform is actually a rotation of , followed by a reflection of the y-axis.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 10:22 pm ad1c9bdddf>
    https://brainmass.com/math/linear-algebra/eigenvectors-eigenvalues-transformation-469484

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