The Rank of a Real Matrix

In the setting of Euclidean Space and Real Matrices: Standing Notation (Algebra) and Linear Algebra: Standing Notation, let $m,n\in\mathbb{N}$, let $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ be a real matrix.

(i) Rank. The rank of $A$ is the dimension of the image of $L_{A}$,

$$\operatorname{rank}A=\dim\operatorname{im}L_{A}$$

(a finite-dimensional subspace of $\mathbb{R}^{m}$ by Real Matrices and Linear Maps between Euclidean Spaces (ii), and a vector space by Linear Subspaces Are Vector Spaces; Intersections (i)).