Vector space/Subspaces of K^123/Example

Let ${}K$ be a field. It is easy to get an overview over the linear subspaces of ${}K^{n}$, as the dimension of a linear subspace equals ${}k$ with ${}0\leq k\leq n$, due to fact. For ${}n=0$, there is only the null space itself; for ${}n=1$, there is the null space and ${}K$ itself. For ${}n=2$, there is the null space, the whole plane ${}K^{2}$, and the one-dimensional lines through the origin. Every line ${}G$ has the form

${}G=Kv={\left\{sv\mid s\in K\right\}}\,,$

with a vector ${}v\neq 0$. Two vectors different from ${}0$ define the same line if and only if they are linearly dependent. For ${}n=3$, there is the null space, the whole space ${}K^{3}$, the one-dimensional lines through the origin, and the two-dimensional planes through the origin.