Matrix-Vector Product

Let $m,n\in\mathbb{N}$. Let $A=(A_{\alpha i})$ be an $m\times n$ matrix with real entries, and let $v=(v_1,\dots,v_n)\in\mathbb{R}^n$. The product vector $Av\in\mathbb{R}^m$ is defined by

for every $\alpha\in\{1,\dots,m\}$.