# linear algebra – The convex hull of a vector space.

For a matrix $$A = (a_1, ldots, a_n) in mathbb{R}^{m times n}$$, denote
$$begin{equation*} | A|_{b, c} := big| (| a_1|_b, ldots, | a_n|_b)^{top} big|_c. end{equation*}$$
Now suppose $$V subseteq mathbb{R}^n$$ and $$A in mathbb{R}^{n times m}$$, does the following relationship hold?
$$begin{equation*} big{ A^{top} b : b in V big} = big{ | A^{top} |_{1, infty} b : b in operatorname{conv} (-V cup V)big}, end{equation*}$$
where $$operatorname{conv}$$ is the convex hull of a set.