Linear subspace/Realization as kernel of a linear mapping/Exercise

Let ${}K$ be a field, and let ${}V$ denote a finite-dimensional ${}K$-vector space. Let ${}U\subseteq V$ be a linear subspace. Show that there exists a ${}K$-vector space ${}W$ and a surjective ${}K$-linear mapping

$\varphi \colon V\longrightarrow W$

such that ${}U=\operatorname {kern} \varphi$ holds.

Create a solution