Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $m,n,p,q\in\mathbb{N}$, $A,A'\in\mathcal{M}_{m\times n}(\mathbb{R})$, $B,B'\in\mathcal{M}_{n\times p}(\mathbb{R})$, $C\in\mathcal{M}_{p\times q}(\mathbb{R})$, $s\in\mathbb{R}$, $x,y\in\mathbb{R}^{n}$, $z\in\mathbb{R}^{p}$ and $w\in\mathbb{R}^{m}$, with real matrices, their entries, entrywise operations, the matrix product, the matrix-vector product, the transpose, the identity matrices $I_{m}$, $I_{n}$ and the standard basis vectors $e_{k}$ of $\mathbb{R}^{n}$, the operations and origin $0$ of $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$, the dot product, the Euclidean norm $\lVert\cdot\rVert$, and sums and squares as in Commutative Rings, Fields and Ordered Fields: Standard Notation §rings.

¶ $(AB)C=A(BC)$.

¶ $A(B+B')=AB+AB'$ and $(A+A')B=AB+A'B$.

¶ $(sA)B=s(AB)=A(sB)$.

¶ $I_{m}A=A=AI_{n}$.

¶ $A(x+y)=Ax+Ay$, $A(x-y)=Ax-Ay$, $A(sx)=s(Ax)$, $(A+A')x=Ax+A'x$, $(sA)x=s(Ax)$, $(AB)z=A(Bz)$, $I_{n}x=x$ and $A0=0$.

¶ For every $k\in[n]$, $(Ae_{k})_{i}=A_{ik}$ for all $i\in[m]$.

¶ $(A^{\top})^{\top}=A$, $(A+A')^{\top}=A^{\top}+A'^{\top}$, $(sA)^{\top}=sA^{\top}$, $(AB)^{\top}=B^{\top}A^{\top}$ and $I_{n}^{\top}=I_{n}$.

¶ $(Ax)\cdot w=x\cdot(A^{\top}w)$.

¶ $\lVert Ax\rVert^{2}\le\Bigl(\sum_{i=1}^{m}\sum_{j=1}^{n}A_{ij}^{2}\Bigr)\lVert x\rVert^{2}$.