Proving Linear Independence in composition of functions

Question

Let $K$ be a field, where $V$ is vector space over $K$ and $g: V \to V$ and $\exists n \in \mathbb N$ such that $g^{(N)}=0$ while $g^{(N-1)}\neq0$. For $v \in V$ with $g^{(N-1)}(v)\neq0$, show that $v, g(v),...,g^{(N-1)}(v)$ are linearly independent.

I am stuck on this question. My thoughts so far:

Since $g^{(N-1)}(v)\neq 0$ that means that $g^{(k)}(v)\neq 0 \forall k \in {0,...,N-1}$, whereby $k \in \mathbb N$. How did this help in the context of proving they're linearly independent?

Answer 1

Hint

$$a_1g(v)+a_2g^{(2)}(v)+...+a_{N-1}g^{(N-1)}(v)=0$$

Now apply, $g^{(N-2)}$ and get:

$$a_1g^{N-1}(v)=0\to a_1=0$$

Use the same idea to get $a_2=a_3=...=a_{N-1}=0$.

Can you finish?