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

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

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

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

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

c) Show that every linear subspace ${}U\subseteq K^{n}$ is the solution space of a system of linear equations.

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