Is $\operatorname{End}_K(V)$ self-opposite? Let $V$ be a vector space over a field $K$. 
Is $\operatorname{End}_K(V)$ a self-opposite $K$-algebra?

If "self opposite" means "isomorphic to its opposite ring," then no.
If you take an infinite dimensional $V$, then it is known that the ring of linear transformations written on the left is right self-injective but not left self-injective. So it cannot be isomorphic to its opposite ring.
In the finite dimensional case, the answer is yes, of course, by using the transposition map.
