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 there exist

linear forms ${}L_{1},\ldots ,L_{r}$ on ${}V$ such that

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

Every linear subspace ${}U\subseteq V$ is the kernel of a linear mapping and every linear subspace of ${}K^{n}$ is the solution space of a linear system.

Write a proof