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Let V be an n-dimensional complex vector space and let $E\in \mbox{End}(V)$ satisfy $E^2 = E$. Show that there is an $r \geq 0$ and a basis$(v_1, v_2, ..., v_n)$ for $V$ such that $E(v_i) = \begin{cases} v_i & \text{if } 1 \leq i \leq r \\ 0 & \text{if } r + 1 \leq i \leq n \end{cases}$

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    $\begingroup$ Be careful with your writing: is $\;T=E\;$ ? $\endgroup$ – DonAntonio Apr 16 '16 at 17:36
  • $\begingroup$ @Joanpemo Thanks, I didn't notice. $\endgroup$ – Dheeraj Putta Apr 16 '16 at 18:00
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Hint:

Show $V=\ker T\oplus\operatorname{Im}T$.

Note this is true for a vector space over any field, not merely $\mathbf C$or $\mathbf R$ .

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  • $\begingroup$ I have proven your hint. Is this all I need to do or is there more? $\endgroup$ – Dheeraj Putta Apr 16 '16 at 18:36
  • $\begingroup$ Only some words to conclude, if you didn't do it. $E$ is the projection onto the second factor. $\endgroup$ – Bernard Apr 16 '16 at 18:40

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