A linear map $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ in Euclidean coordinates

Let $m,n\in \mathbb{N}$, including zero. A map $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ is linear when

$L(\alpha u+\beta v)=\alpha L(u)+\beta L(v)$

for all $u,v\in {\mathbb{R}}^m$ and all $\alpha ,\beta \in \mathbb{R}$. Both spaces carry their Euclidean vector-space operations and Euclidean norms from The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$.

This is precisely the identity in Linear map between vector spaces over the same field, applied to the real vector spaces ${\mathbb{R}}^m$ and ${\mathbb{R}}^n$ with their pointwise operations; thus the abstract and coordinate notions agree. If $m=0$, linearity forces the sole domain point $0$ to map to $0$, and this unique map is linear. If $n=0$, the unique map into the zero space is linear. These assertions use the zero-space and norm conventions of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, not a maximum over an empty coordinate set.

This is the concrete Euclidean notion required for total differentiation. It makes no assertion about linear maps between arbitrary vector spaces.