# Show {$I,N,N^2$} forms a basis of V iff $N^2 \neq 0$

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

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If $a_1 I + a_2 N + a_3 N^2 = 0$, multiply by $N^2$ and use $N^3=0$ to get $a_1 = 0$, then ...

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