Let $V$ be a vector space over a field $F$ and $T$ a linear operator on $V$. If $T^2$$=$ $0$, what can you say about the relation of the range of $T$ to the null space of $T$?
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We can say:
$$T^2=0\iff \operatorname{Im} T\subset\ker T$$ and the proof is pretty easy.
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Hints:
$$T^2=0\implies \text{Im}\,T\subset\ker T\implies \dim\text{Im}\,T\le\dim\ker T\ldots$$
DonAntonio
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why Im T is a subset of Ker T? – Topology May 26 '14 at 18:04
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@Mathematics: $$y\in\text{Im},T\implies \exists,x;;s.t.;;y=Tx\implies Ty=T^2x=0\ldots$$ – DonAntonio May 26 '14 at 18:19
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oh! Yeah! thank you! @ DonAntonio – Topology May 26 '14 at 18:32