given $\displaystyle K$ a field. Find all the linear subspaces of the $\displaystyle K$-vector space $\displaystyle K^2= \{(x,y) \colon x,y \in K \} \text{for }K=\mathbf{F_2}\text{ and } K=\mathbf{F_3}$.
This seems too easy, so here is what I did, please correct me.
$\displaystyle U$ is a linear subspace of $\displaystyle K^2$ if and only if
1)$\displaystyle U \neq \emptyset$.
2)$\displaystyle \alpha v+\beta u \in U, \forall \alpha , \beta \in K \text{ and } \forall v,u \in U \Rightarrow 0 \in U$.
So $\displaystyle \mathbf{F_1}^2 \text{ with } 0 \in \mathbf{F_1}$ is one, $\displaystyle \mathbf{F_2}^2$ is the other one for $\displaystyle K= \mathbf{F_2}$.
and it's pretty much the same thing for $\displaystyle K=F_3$, except that $\displaystyle {F_3}^2$ is also one.
Thanks in advance.