I came across a problem that I would like to ask you about:
Let $N \in Mat_{nxn}(K)$ i.e square matrix, such that $N^{3}=0$, and $A=\lambda I +N$, where $\lambda \in K$.
Also, $V$ is a vector space $V=span(I,N,N^{2},N^3,N^4,...)$
I found this set $B=${$I,N,N^2$} to be a generating set of all $V$ since all powers are spanned by B.
now I need to prove that B forms a basis if and only if $N^2 \neq 0$.
I.E. linear independence needs to be shown, right?: $a_1I+a_2N+a_3N^2=0 \implies a_1=a_2=a_3=0$
I'm a bit stuck right now and don't see how to proceed from here, since I don't know much about N. Any hints or tips would be greatly appreciated!.
Best