Rank of a Real Matrix: Rank–Nullity, Invertibility and the Size Bound

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 (so $L_{A}$ is linear by Real Matrices and Linear Maps between Euclidean Spaces (i)), and let $\operatorname{rank}A$ be the rank of $A$.

(i) Rank–nullity. $\ker L_{A}$ is a finite-dimensional subspace of $\mathbb{R}^{n}$, and

$$\operatorname{rank}A+\dim\ker L_{A}=n.$$

(ii) Invertibility. If $m=n$, then $A$ is invertible if and only if $\operatorname{rank}A=n$.

(iii) Size bound.

$$\operatorname{rank}A\le\min(m,n).$$