Maps Formed Pointwise from Vector and Matrix Operations

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $S$ be a set and let $p,q,r\in\mathbb{N}$. Sums, scalar multiples, negatives and differences of points of $\mathbb{R}^{q}$ and of matrices are those of Euclidean Space and Real Matrices: Standing Notation (ii) and Euclidean Space and Real Matrices: Standing Notation (iii); the matrix product, the matrix-vector product and the transpose are those of Euclidean Space and Real Matrices: Standing Notation (iii); and invertible matrices and their inverses are as in Euclidean Space and Real Matrices: Standing Notation (x). Let $f,g:S\to\mathcal{M}_{p\times q}(\mathbb{R})$, $h:S\to\mathcal{M}_{q\times r}(\mathbb{R})$, $x,y:S\to\mathbb{R}^{q}$ and $\varphi:S\to\mathbb{R}$ be maps.

(i) There is exactly one map $S\to\mathcal{M}_{p\times q}(\mathbb{R})$ whose value at every $u\in S$ is $f(u)+g(u)$, and likewise exactly one such map with values $\varphi(u)f(u)$, exactly one with values $-f(u)$ and exactly one with values $f(u)-g(u)$. There is exactly one map $S\to\mathcal{M}_{p\times r}(\mathbb{R})$ with values $f(u)h(u)$, exactly one map $S\to\mathbb{R}^{p}$ with values $f(u)x(u)$, and exactly one map $S\to\mathcal{M}_{q\times p}(\mathbb{R})$ with values $f(u)^{\top}$. There is exactly one map $S\to\mathbb{R}^{q}$ with values $x(u)+y(u)$, and likewise exactly one with values $\varphi(u)x(u)$, exactly one with values $-x(u)$ and exactly one with values $x(u)-y(u)$. These maps are written $u\mapsto f(u)+g(u)$, $u\mapsto\varphi(u)f(u)$, and so on.

(ii) If $p=q$ and $f(u)$ is invertible for every $u\in S$, then there is exactly one map $S\to\mathcal{M}_{p}(\mathbb{R})$ whose value at every $u\in S$ is $f(u)^{-1}$; it is written $u\mapsto f(u)^{-1}$.