Exercise I.B.4

Question

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ewcommand{\Hom}[2]{\mathrm{Hom}\left( #1, #2 ight)}$
Let $V$ be a vector space, and let $ho:V 	o V$ be an idempotent vector endomorphism. Show that $ho$ acts as the identity on $\Hom{ho(V)}{ho(V)}$.

Toğrul Topal · Accepted Answer

By associativity we have
$$ (f \circ \rho) \circ \rho = f \circ (\rho \circ \rho) = f \circ \rho$$
But $f\circ \rho (\rho(x)) = f (\rho (x))$ for all $x \in V$ implies $f \circ \rho = f$, and so we have the right multiplicative identity. The left multiplicative identity of $\rho$ follows similarly.

Exercise I.B.4

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Exercise I.B.4

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