Matrix-Vector Product

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $m,n\in\mathbb{N}$, let $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ be a real matrix with entries $A_{ij}$, and let $x\in\mathbb{R}^{n}$, Euclidean space, with components $x_{j}$ as in Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii).

(i) The matrix-vector product $Ax\in\mathbb{R}^{m}$ is the value at $(A,x)$ of the map of The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique (ii), so that

$$(Ax)_{i}=\sum_{j=1}^{n}A_{ij}x_{j}\qquad(i\in[m]).$$