Linear forms/Relation between kernels/Linear subspaces/Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space over a field ${}K$, and let ${}L,L_{1},\ldots ,L_{m}$ denote linear forms on ${}V$. Show that the relation

${}\bigcap _{i=1}^{m}\operatorname {kern} L_{i}\subseteq \operatorname {kern} L\,$

holds if and only if ${}L$ belongs to the linear subspace (in the dual space) generated by the ${}L_{1},\ldots ,L_{m}$.

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