Proof of Maps Formed Pointwise from Vector and Matrix Operations

Proof of (i). For a set $J$, let $\oplus_{J}$ be the binary operation $+$ on $\mathbb{R}^{J}$ of Pointwise Operations on Real-Valued Maps Exist and Are Unique (i), a map $\mathbb{R}^{J}\times\mathbb{R}^{J}\to\mathbb{R}^{J}$, and let $\mu_{J}:\mathbb{R}\times\mathbb{R}^{J}\to\mathbb{R}^{J}$ and $\nu_{J}:\mathbb{R}^{J}\to\mathbb{R}^{J}$ be the maps of Pointwise Operations on Real-Valued Maps Exist and Are Unique (ii) and Pointwise Operations on Real-Valued Maps Exist and Are Unique (iii). By Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (i), $\mathcal{M}_{p\times q}(\mathbb{R})=\mathbb{R}^{J}$ for $J=[p]\times[q]$, and by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv) and Pointwise Operations on Real-Valued Maps (i), for $A,B\in\mathcal{M}_{p\times q}(\mathbb{R})$ and $t\in\mathbb{R}$,

$$A+B=\oplus_{J}(A,B),\qquad tA=\mu_{J}(t,A),\qquad -A=\nu_{J}(A),\qquad A-B=A+(-B)=\oplus_{J}(A,\nu_{J}(B)).$$

By Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (i), $\mathbb{R}^{q}=\mathbb{R}^{[q]}$, and by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv) and Pointwise Operations on Real-Valued Maps (i) the same four formulas hold for points of $\mathbb{R}^{q}$, with $J=[q]$.

Sums, scalar multiples, negatives and differences. By Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), with $\oplus_{J}$ for $F$, there is exactly one map $S\to\mathcal{M}_{p\times q}(\mathbb{R})$ with value $\oplus_{J}(f(u),g(u))=f(u)+g(u)$ at every $u\in S$; with $\mu_{J}$ for $F$, $\varphi$ for its $f$ and $f$ for its $g$, exactly one with value $\mu_{J}(\varphi(u),f(u))=\varphi(u)f(u)$. By the second sentence of Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), with $\nu_{J}$ for $\Phi$, the composite $\nu_{J}\circ f$ is the only map $S\to\mathcal{M}_{p\times q}(\mathbb{R})$ with value $\nu_{J}(f(u))=-f(u)$ at every $u$; likewise $\nu_{J}\circ g$ has value $-g(u)$. By Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii) with $\oplus_{J}$ for $F$, $f$ for its $f$ and $\nu_{J}\circ g$ for its $g$, there is exactly one map with value $\oplus_{J}(f(u),\nu_{J}(g(u)))=f(u)-g(u)$ at every $u$. The same arguments, with $J=[q]$ and $x$, $y$ in place of $f$, $g$, give the four maps into $\mathbb{R}^{q}$.

Products, matrix-vector products and transposes. Let $\pi:\mathcal{M}_{p\times q}(\mathbb{R})\times\mathcal{M}_{q\times r}(\mathbb{R})\to\mathcal{M}_{p\times r}(\mathbb{R})$, $\phi:\mathcal{M}_{p\times q}(\mathbb{R})\times\mathbb{R}^{q}\to\mathbb{R}^{p}$ and $\tau:\mathcal{M}_{p\times q}(\mathbb{R})\to\mathcal{M}_{q\times p}(\mathbb{R})$ be the maps of The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique (i), The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique (ii) and The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique (iii), with $p,q,r$ for $m,n,p$ in the first and $p,q$ for $m,n$ in the other two. By Product of Real Matrices (i), Matrix-Vector Product (i) and Transpose of a Real Matrix (i), $\pi(A,C)=AC$, $\phi(A,z)=Az$ and $\tau(A)=A^{\top}$ for all $A\in\mathcal{M}_{p\times q}(\mathbb{R})$, $C\in\mathcal{M}_{q\times r}(\mathbb{R})$ and $z\in\mathbb{R}^{q}$. So Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), with $\pi$ for $F$, $f$ and $h$ for its $f$ and $g$, gives exactly one map $S\to\mathcal{M}_{p\times r}(\mathbb{R})$ with values $f(u)h(u)$; with $\phi$ for $F$, $f$ and $x$ for its $f$ and $g$, exactly one map $S\to\mathbb{R}^{p}$ with values $f(u)x(u)$; and its second sentence, with $\tau$ for $\Phi$, shows that $\tau\circ f$ is the only map $S\to\mathcal{M}_{q\times p}(\mathbb{R})$ with values $f(u)^{\top}$.

Proof of (ii). Let $\mathrm{GL}_{p}(\mathbb{R})$ be the set of invertible matrices in $\mathcal{M}_{p}(\mathbb{R})$ of The Invertible Matrices and the Positive Semidefinite Matrices Form Sets (i), and let $\iota:\mathrm{GL}_{p}(\mathbb{R})\to\mathcal{M}_{p}(\mathbb{R})$ be the map $B\mapsto B^{-1}$ of Perturbations of an Invertible Matrix: the Invertible Matrices Form an Open Set and Inversion Is Continuous (iv). Since $f(u)\in\mathrm{GL}_{p}(\mathbb{R})$ for every $u\in S$, by The Invertible Matrices and the Positive Semidefinite Matrices Form Sets (i), $f$ is also a map $S\to\mathrm{GL}_{p}(\mathbb{R})$, by Functions, Values of a Function, and Functions from One Class to Another (iii) and Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (v). By the second sentence of Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), with $\mathrm{GL}_{p}(\mathbb{R})$ for $Y$ and $\iota$ for $\Phi$, the composite $\iota\circ f$ is the only map $S\to\mathcal{M}_{p}(\mathbb{R})$ with value $\iota(f(u))=f(u)^{-1}$ at every $u\in S$.