What is $D$, and what is Z(v,f)?
I have a question on cyclic subspaces.
V is a finite dimensional space over the field F. There exists f:V->V
It's given to me that , k is minimal, such that the set $(v,f(v),f^2(v),...,f^k(v))$ is dependent.
Now I have to prove that
1-)$D=(v,f(v),f^2(v),...,f^(k-1))$
2-)D spans Z(v,f)
I proved the first one which was very easy. But I am having a problem proving that it spans the cyclic subspace... Can someone give me a direction. Thanks
Sorry, I didn't write it very clear.
The first one claims that D is independent
Z(v,f) is cyclic subspace . in other words,
Definition: V is a vector space over the field F. f:V->V is a linear operator. $v\inV$ , then
$Z(v,f)=Span(v,f(v),f^2(v),...,f^i(v),...)$ is cyclic subspace of v on f
The first claim is pretty easy , since we know k is minimal, then we would have that D is independent.
But the second one I couldn't prove.