I'm in trouble with an exercise. The text asks to consider a quiver with only one loop, a field $k = \mathbb{F}_2(t)$, and a representation $V$ given by the matrix: \begin{bmatrix} 0 & t \\ 1 & 0 \end{bmatrix} It asks to calculate the endomorphism ring, $End_k(V)$ and $End_\bar{k}(V)$, where $\bar{k}$ is the algebraic closure of k.
For the ring $End_k(V)$ I applied the definition of morphism and I found that the ring is the set of matrices: \begin{bmatrix} a & ct \\ c & a \end{bmatrix} with $a$ and $c$ in $k$.
What should I do now for the case with $\bar{k}$?