Intuition/motivation behind t-cyclic subspaces

Question

I am having trouble understanding $T$-cyclic subspaces. My textbook gives the following definition:

Let $T$ be a linear operator on a vector space $V$, and let $x$ be a nonzero vector in $V$. The subspace: $$W = span([x,T(x),T^2(x),...])$$ is called the $T$-cyclic subspace of $V$ generated by $x$.

I am having trouble understanding what this means and the motivation behind such a definition. Once again in linear algebra I find myself wondering: "who cares?" If anyone could provide any intuition or motivation behind this definition that would be awesome. Or even any kind of application of this definition.

Thank you!

Answer 1

The cyclic subspaces are very useful for one thing: they are subspaces that are invariant with respect to $T$! That means if $x\in W$, $f(T)x \in W$, where $f$ is a polynomial.

Other useful examples of T-invariant subspaces include $KerT$, $ImT$. Hope this helps!