I am currently learning Linear Algebra and just can't finish this one exercise, at least partially...
We are given some field $F$ and the normal vector space $V = K^n$. By $Z(A,v)$ we denote $\left\langle v, Av, A^2v, A^3v,\dots\right\rangle$ where $A$ is just some $n \times n$ matrix. As an added condition we are given that $m_v$ and $m_w$ are coprime with $m_v$ being the normalised polynomial of lowest possible degree such that $m_v(A)v = 0$.
Now the actual task is to show that $Z(A,v) \oplus Z(A, w) = Z(A, v+w)$.
I was already able to show the inclusion $Z(A,v) + Z(A, w) \supseteq Z(A, v+w)$, which was pretty easy and to show that $Z(A, v) \cap Z(A,w) = \{0\}$ I used that $m_v$ and $m_w$ are coprime.
I tried approaching the other inclusion of $Z(A,v) \oplus Z(A, w) \subseteq Z(A, v+w)$ similary by choosing some $x \in Z(A,v) \oplus Z(A, w)$. Then we know that there is a single and clear way to write that as $$x = \sum_{n \in \mathbb{N}} A^n(\lambda_nv+\mu_nv).$$
And at this point I dont see how I could show $\sum_{n \in \mathbb{N}} A^n(\lambda_nv+\mu_nv) \in Z(A, v+w) = \left\langle v+w, A(v+w), A^2(v+w), A^3v,\dots\right\rangle$ as the left hand side seems way more general than the right hand one. I guess I have to use that $m_v$ and $m_w$ are coprime somewhere here maybe, but I lack a concrete idea...
Already thanks for any help in advance! :)
The general element of $Z(A, v) + Z(A, w)$ is of the form $p(A)v + q(A)w$, where $p$ and $q$ are polynomials which matter up to multiples of $m_v$ and $m_w$ respectively. We wish to show that this is of the form $r(A)(v+w)$ for some polynomial $r$. Observe that $r(A)(v+w) = r(A)v + r(A)w$. Thus, we wish to find some polynomial $r$ such that $r \equiv p \bmod{m_v}$ and $r \equiv q \bmod{m_w}$. Do you see how to proceed?