Real Matrices and Linear Maps between Euclidean Spaces

In the setting of Euclidean Space and Real Matrices: Standing Notation (Algebra) and Linear Algebra: Standing Notation, let $m,n,p\in\mathbb{N}$, with $\mathbb{R}^{n}$, $\mathbb{R}^{m}$ and $\mathbb{R}^{p}$ the real vector spaces of tuples.

(i) Linearity. For every $A\in\mathcal{M}_{m\times n}(\mathbb{R})$, $L_{A}$ is a linear map from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$.

(ii) Kernel and image. For every $A\in\mathcal{M}_{m\times n}(\mathbb{R})$, $\ker L_{A}$ is a linear subspace of $\mathbb{R}^{n}$, and $\operatorname{im}L_{A}$ is a finite-dimensional linear subspace of $\mathbb{R}^{m}$.

(iii) Representation. For every linear map $T$ from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ there is exactly one $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ with $T=L_{A}$; its $j$-th column is $Te_{j}$, that is,

$$A_{ij}=(Te_{j})_{i}\qquad(i\in[m],\ j\in[n]).$$

(iv) Products. For $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ and $B\in\mathcal{M}_{n\times p}(\mathbb{R})$, $L_{AB}$ is the composite of $L_{B}$ followed by $L_{A}$:

$$L_{AB}=L_{A}L_{B}.$$

(v) Invertibility. For a square matrix $A\in\mathcal{M}_{n}(\mathbb{R})$ the following are equivalent: $A$ is invertible; $L_{A}$ is bijective; $\ker L_{A}=\{0\}$.