Linear and Multilinear Algebra

Let $e_1,e_2,\dots,e_n$ be the standard basis vectors of $\mathbb{R}^n$. Let $V$ be a subspace of $\mathbb{R}^n$, and for each $i$, choose a vector $v_i\in V$ such that the Euclidean distance between $v_i$ and $e_i$ is as small as possible.