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Let [itex]T : V \to V[/itex] be a linear operator on an n-dimensional vector space V. Assume that the characteristic polynomial of T splits over F (the field underlying V). Prove that the following are equivalent:
- There exists a vector [itex]x \in V[/itex] such that [itex]\{x,T(x),T^2(x),\dots ,T^{n - 1}(x)\}[/itex] is linearly independent.
- The characteristic polynomial of T is equal to [itex](-1)^n[/itex] times the minimal polynomial of T.
- There exists a basis [itex]\beta[/itex] such that
[tex][T]_{\beta} = \left (\begin{array}{ccccc}0 & 0 & \dots & 0 & (-1)^{n - 1}a_0\\1 & 0 & \dots & 0 & (-1)^{n - 1}a_1\\0 & 1 & \dots & 0 & (-1)^{n - 1}a_2\\ \vdots & \vdots & \ddots & \vdots & \vdots \\0 & 0 & \dots & 1 & (-1)^{n - 1}a_{n - 1}\end{array}\right )[/tex]
where [itex](-1)^nt^n + a_{n - 1}t^{n - 1} + \dots + a_1t + a_0[/itex] is the characteristic polynomial of T.
(Hint: For proving that (2) implies (3), it is helpful to show that the Jordan canonical form of T is the same as the Jordan canonical form of the matrix given in part (3)).
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