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Math Help - matrix for linear transformation

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    matrix for linear transformation

    Last question ever, I promise.

    If T is a linear operator on the space V and the eigenvector of T is a non-zero vector v in V such that Tv=av for some scalar a. Suppose that V is finite-dimensional and that B is an ordered basis of V such that the first vector v is an eigenvector and a is the corresponding eigenvalue. Show that the first column of [T]B has a as its first entry and the rest of those entries are zero.

    I have absolutely no idea what to do here.

    Any help would be appreciated.
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  2. #2
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    Quote Originally Posted by PvtBillPilgrim View Post
    Last question ever, I promise.

    If T is a linear operator on the space V and the eigenvector of T is a non-zero vector v in V such that Tv=av for some scalar a. Suppose that V is finite-dimensional and that B is an ordered basis of V such that the first vector v is an eigenvector and a is the corresponding eigenvalue. Show that the first column of [T]B has a as its first entry and the rest of those entries are zero.

    I have absolutely no idea what to do here.

    Any help would be appreciated.
    This is simple.

    We are given a linear operator
    T:V\mapsto V
    Where V is n-dimensional real vector space.
    Thus, there is a n\times n standard matrix corresponding to the transformation.

    Let B=(\b{u}_1,...,\b{u}_n) be the ordered basis.

    We are told that the first vector \b{u}_1 is an eigenvector of the transformation. That is,
    T(\b{u}_1)=a\b{u}_1
    Now relative to the ordered basis, its coordinate vector is,
    [T(\b{u}_1)]_B=(a,0,...,0)
    Because, a\b{u}_1=a\b{u}_1+0\b{u}_2+...+0\b{u}_n

    Finally, the standard matrix for the linear operator is given by,
    [T]_B=\big[[T(\b{u}_1)]_B\big|[T(\b{u}_2)]_B\big|....\big|[T(\b{u}_n)]_B\big]
    But the first coloum in this matrix is,
    [T(\b{u}_1)]_B
    The coordinate vector relative to the ordered basis.
    Which we found to be,
    \left[\begin{array}{c}a\\0\\0\\...\\0 \end{array} \right]
    Thus the first entry is a and everything else is zero.
    Last edited by ThePerfectHacker; February 1st 2007 at 02:29 PM.
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