Lines, Scalar Multiples, Basis Directions, Small Balls and Matrix Rows in ℝⁿ

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $n\in\mathbb{N}$, $x,y,h,w\in\mathbb{R}^{n}$, $s,t\in\mathbb{R}$ and $i\in[n]$, with the operations of Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv) on $\mathbb{R}^{n}$, the Euclidean norm $\lVert\cdot\rVert$, the dot product $\cdot$, the Euclidean distance $d_{E}$ and the $i$-th standard basis vector $e_{i}$.

(i) Zero multiple. $x+0h=x$.

(ii) Unit multiple. $1h=h$.

(iii) Moving along a line.

$$(x+sh)+th=x+(s+t)h .$$

(iv) Differences along a line.

$$(x+sh)-(x+th)=(s-t)h .$$

(v) Multiples of differences. $t(x-y)=tx-ty$.

(vi) Segments as lines.

$$(1-t)x+t(x+h)=x+th .$$

(vii) Length of a basis multiple. $\lVert te_{i}\rVert=|t|$.

(viii) Dot product with a basis multiple. $w\cdot(te_{i})=t\,w_{i}$.

(ix) Distance along a line.

$$d_{E}(x,x+th)=|t|\,\lVert h\rVert .$$

(x) Small displacements stay in an open set. If $U\subseteq\mathbb{R}^{n}$ is open and $a\in U$, there is a real $r>0$ such that $a+v\in U$ for every $v\in\mathbb{R}^{n}$ with $\lVert v\rVert<r$.

(xi) Small steps along a coordinate axis. If $U\subseteq\mathbb{R}^{n}$ is open and $a\in U$, there is a real $r>0$ such that $a+ue_{i}\in U$ for every real $u$ with $|u|<r$.

(xii) Coordinates of a matrix-vector product. Let $m\in\mathbb{N}$, let $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ and let $k\in[m]$. There is a point $\rho\in\mathbb{R}^{n}$ with $\rho_{j}=A_{kj}$ for every $j\in[n]$, and every such $\rho$ satisfies, for the matrix-vector product,

$$(Av)_{k}=\rho\cdot v\qquad\text{for every }v\in\mathbb{R}^{n}.$$