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Math Help - Quick basis question

  1. #1
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    Quick basis question

    Suppose V is a vector space of dimension n, that T is a linear transformation on V and that there exists v \in V such that \{v, Tv, ..., T^{n-1}(v)\} is a basis of V.

    Then clearly T^{n}(v) can be written as a linear combination of the basis elements, but can we find this linear combination explicitly? I suspect not...
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    Quote Originally Posted by Amanda1990 View Post
    Suppose V is a vector space of dimension n, that T is a linear transformation on V and that there exists v \in V such that \{v, Tv, ..., T^{n-1}(v)\} is a basis of V.
    So that T has rank n and is an invertible matrix.

    Then clearly T^{n}(v) can be written as a linear combination of the basis elements, but can we find this linear combination explicitly? I suspect not...
    This would depend on the "characteristic equation" of T, [tex]|T- \lambda I|= 0[/itex] Since every linear transformation satisfies its own characteristic equation, that gives an n^{th} degree polynomial for T and that can be solved for T^n. That's not quite what you are asking but it's the best I can think of.
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  3. #3
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    Quote Originally Posted by Amanda1990 View Post
    Suppose V is a vector space of dimension n, that T is a linear transformation on V and that there exists v \in V such that \{v, Tv, ..., T^{n-1}(v)\} is a basis of V.

    Then clearly T^{n}(v) can be written as a linear combination of the basis elements, but can we find this linear combination explicitly? I suspect not...
    Here's a simple example, with n=3. Let T be the linear transformation of \mathbb{R}^3 given by the matrix \begin{bmatrix}0&0&\alpha\\ 1&0&\beta\\ 0&1&\gamma\end{bmatrix}. Let e_1,\;e_2,\;e_3 be the vectors \begin{bmatrix}1\\0\\0\end{bmatrix}, \begin{bmatrix}0\\1\\0\end{bmatrix}, \begin{bmatrix}0\\0\\1\end{bmatrix} in the standard basis. If v = e_1 then Tv=e_2 and T^2v=e_3. So \{v, Tv,T^2(v)\} is a basis. But T^3v = Te_3 = \begin{bmatrix}\alpha\\\beta\\\gamma\end{bmatrix}, which is an arbitrary vector. So you are right to think that there is no explicit way to determine it.
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