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 $L_{1}, \dots, L_{r}$ on $V$ such that

U = ⋂_{i = 1}^{r} 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