Linear subspace/Endomorphisms with invariant space/Ring/Dimension/Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space, and ${}U\subseteq V$ a linear subspace. Show that

${}R(U)={\left\{\varphi \in \operatorname {End} _{}{\left(V\right)}\mid U{\text{ is a }}\varphi -{\text{invariant linear subspace}}\right\}}\,$

is, with the natural addition and multiplication of endomorphisms, a ring, and a linear subspace of ${}\operatorname {End} _{}{\left(V\right)}$. Determine the dimension of this space.

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