given K a field. Find all the linear subspaces of the K-vector space 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.
U is a linear subspace of K^2 if and only if
1) U \neq \emptyset.
2) \alpha v+\beta u \in U, \forall \alpha , \beta \in K \text{ and } \forall v,u \in U \Rightarrow 0 \in U.
So \mathbf{F_1}^2 \text{ with } 0 \in \mathbf{F_1} is one, \mathbf{F_2}^2 is the other one for K= \mathbf{F_2}.
and it's pretty much the same thing for K=F_3, except that {F_3}^2 is also one.
Thanks in advance.