Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $p,q\in\mathbb{N}$, and let $X,Y,Z,X',Y'$ be all in $\mathbb{R}^{q}$ or all in $\mathcal{M}_{p\times q}(\mathbb{R})$. Sums, negatives and differences are those of Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv) in the first case and of Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv) in the second, and $0$ is the origin of $\mathbb{R}^{q}$ in the first case and the zero matrix $0_{p\times q}$ in the second.

(i) $X+Y=Y+X$, $(X+Y)+Z=X+(Y+Z)$, $X+0=X=0+X$ and $X+(-X)=0$.

(ii) $X-X=0$, $X-0=X$, $0-X=-X$, $-(-X)=X$, $X-Y=-(Y-X)$, $X-Y=X+(-Y)$, $(X+Y)-(X'+Y')=(X-X')+(Y-Y')$, $(X-Y)+Y=X$ and $(X-Y)+(Y-Z)=X-Z$; and if $X-Y=0$, then $X=Y$.