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Let V be a vector space over the field K. Show that $(\text{End}_K(V),+,\circ)$ is a ring.

I know that an endomorphism is a homomorphism from V to V. So for example i should show that + is associative with $\forall a,b,c \in \text{End}_K(V) : a+(b+c)=(a+b)+c$ but how do i do that with endomorphisms??

Arji
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  • Do you understand the definition of $+$ in the first line? – Orat May 04 '17 at 18:15
  • @Orat what im not understanding is what objects there are in End$_K(V)$ and how $+$ should work between them. – Arji May 04 '17 at 18:20

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First of all, $\operatorname{End}_K(V)$ is the set consisting of all endomorphisms of $V$, that is, $K$-linear maps from $V$ to $V$.

The addition $+$ in the first line is defined as follows: For $f, g \in \operatorname{End}_K(V)$, the sum $f + g \in \operatorname{End}_K(V)$ is defined by $$ (f + g)(v) = f(v) + g(v) $$ for every $v \in V$. In other words, the sum of endomorphisms is carried over from that of the ground field $K$. Hence associativity, for example, is now a trivial stuff.

Orat
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