# Vector space/Subspaces of K^123/Example

Let ${\displaystyle {}K}$ be a field. It is easy to get an overview over the linear subspaces of ${\displaystyle {}K^{n}}$, as the dimension of a linear subspace equals ${\displaystyle {}k}$ with ${\displaystyle {}0\leq k\leq n}$, due to fact. For ${\displaystyle {}n=0}$, there is only the null space itself, for ${\displaystyle {}n=1}$ there is the null space and ${\displaystyle {}K}$ itself. For ${\displaystyle {}n=2}$, there is the null space, the whole plane ${\displaystyle {}K^{2}}$, and the one-dimensional lines through the origin. Every line ${\displaystyle {}G}$ has the form
${\displaystyle {}G=Kv={\left\{sv\mid s\in K\right\}}\,,}$
with a vector ${\displaystyle {}v\neq 0}$. Two vectors different from ${\displaystyle {}0}$ define the same line if and only if they are linearly dependent. For ${\displaystyle {}n=3}$, there is the null space, the whole space ${\displaystyle {}K^{3}}$, the one-dimensional lines through the origin, and the two-dimensional planes through the origin.