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Let $T: V \rightarrow V$ be a linear transformation that satisfies $T^2=T$.

a) How to prove that if $\lambda$ is an eigenvalue of $T$, then $\lambda=0$ or $\lambda=1$?

b) How to prove that $x - T(x) \in Ker(T)$?

c) How to prove that $V = Ker(T) \oplus Im(T)$?

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closed as off-topic by Cameron Williams, rschwieb, TheGeekGreek, Davide Giraudo, Ken Duna Apr 17 '17 at 19:53

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  • 3
    $\begingroup$ could you please show us your own effort? $\endgroup$ – Arnaldo Apr 17 '17 at 15:09
  • $\begingroup$ Unfortunately, I don't even know where to start. $\endgroup$ – J. Griez Apr 17 '17 at 15:10
  • $\begingroup$ You might start with one of the related questions listed at right, such as math.stackexchange.com/q/1290771/265466. $\endgroup$ – amd Apr 17 '17 at 15:13
  • $\begingroup$ All of these questions have numerous solutions on the site. Even if you're unwilling to post your own work, you should not be so remiss as to look for things that are already there. If you are unwilling to do even that then I don't think anyone will have much sympathy for your questions. $\endgroup$ – rschwieb Apr 17 '17 at 15:23
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Hint 1:

$$T(u)=\lambda u \Rightarrow T^2(u)=\lambda^2 u$$

Now use $T^2=T$.

Hint 2:

$$T(x-T(x))=...$$

Hint 3 $$x=(x-T(x))+T(x)$$

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