Let $T:V\to V$ be a linear transformation of a finite dimensional vector space over a field F to itself. Assume that $$ \{v, Tv, T^2v, \dotsc \} $$ span $V$ for some $v\in V$. Show that

(i) there exists a $k$ such that $v, Tv, \dots , T^{k-1}v $ are linearly independent and $T^kv$ can be written as a linear combination of the assumed spanning set.

(ii) the assumed spanning set forms a basis for $V$.

Find the minimal and characteristic polynomials for $T$.

I sort of know what's going on but I don't quite know how to show it. Any help would be great!


It is well know that every set $W$ that generates a space $V$, then there must be a basis $U\subset W$ for $V$.

It is straightfoward to demonstrate that directly in this exercise defining the set $U$ as the biggest set such that $U=\{v,Tv,...,T^{k-1}v\}$ for some integer $k$ and $U$ is linearly independent.

($k=1$ makes aa lower bound, and if there were no upper bound, $V$ would have infinite dimension, can you see why?)

By forcing these properties in $U$ you can make parts (i) and (ii).

About finding the minimal and characetristic, you should observe a non-trivial combination on the set $U\cup\{T^kv\}$ and infer the polynomial aspect of this combination.

  • 1
    $\begingroup$ thanks I'm still a bit stuck on the minimal and characteristic polynomials though? $\endgroup$ – Tom Oct 25 '13 at 12:37

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