# Let V be an n-dimensional complex vector space and let $T\in \mbox{End}(V)$ satisfy $E^2$ = E

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 $E(v_i) = \begin{cases} v_i & \text{if } 1 \leq i \leq r \\ 0 & \text{if } r + 1 \leq i \leq n \end{cases}$

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

Show $V=\ker T\oplus\operatorname{Im}T$.
Note this is true for a vector space over any field, not merely $\mathbf C$or $\mathbf R$ .
• Only some words to conclude, if you didn't do it. $E$ is the projection onto the second factor. – Bernard Apr 16 '16 at 18:40