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Image spanned by its eigenvectos

  1. May 3, 2009 #1
    Here's the question I'm stuck at:
    Suppose a mapping from R^2 -> R^2 is defined by some 2x2 matrix W. Is the image of W spanned by the eigenvectors of W? Why or why not?

    3. The attempt at a solution
    I know that the eigenvectos of W for a linearly independent set. I also know that the set spanning W will consist of the smallest subspace of W consisting of linear combinations of all vectors in W. But I'm confused about what the eigenvectors actually are. I'm assuming that the answer is "yes", but I don't know why.
     
  2. jcsd
  3. May 4, 2009 #2

    HallsofIvy

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    Suppose
    [tex]W= \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}[/tex]
    Then all eigenvectors or W are multiples of
    [tex]\begin{bmatrix}0 \\ 1\end{bmatrix}[/tex]
    but the image of W is all or R2.
     
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