Finite-dimensional vector space/Linear subspace/Kernel/Solution space/Fact

Let ${}V$ be a finite-dimensional ${}K$-vector space and ${}U\subseteq V$ a linear subspace. Then the following hold.

a) There exist linear forms ${}L_{1},\ldots ,L_{r}$ on ${}V$ such that

${}U=\bigcap _{i=1}^{r}\operatorname {kern} L_{i}\,.$

b) ${}U\subseteq V$ is the kernel of a linear mapping on ${}V$ to some ${}K^{r}$.

c) Every linear subspace ${}U\subseteq K^{n}$ is the solution space of a

system of linear equations.

Write a proof