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

Identities along lines. By Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iii), two points of $\mathbb{R}^{n}$ are equal when their $k$-th components agree for every $k\in[n]$, and by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv) the $k$-th components of the two sides of (ii), (iii), (iv), (v) and (vi) are

$$1h_{k}=h_{k},\qquad (x_{k}+sh_{k})+th_{k}=x_{k}+(s+t)h_{k},\qquad (x_{k}+sh_{k})-(x_{k}+th_{k})=(s-t)h_{k},$$ $$t(x_{k}-y_{k})=tx_{k}-ty_{k},\qquad (1-t)x_{k}+t(x_{k}+h_{k})=x_{k}+th_{k},$$

identities of real arithmetic. For (i), $0h=0$ by Rules for Pointwise Operations on Real-Valued Maps: a Commutative Ring, Scalar Multiples and Evaluation (iv), and $x+0=x$ by Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i).

Basis directions and distances. By Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (v) and Standard Basis Vectors of ℝ^q: Coordinates, Orthonormality and Unit Length (iii),

$$\lVert te_{i}\rVert=|t|\,\lVert e_{i}\rVert=|t|,$$

which is (vii). By The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite (iii), The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite (i) and Standard Basis Vectors of ℝ^q: Coordinates, Orthonormality and Unit Length (i),

$$w\cdot(te_{i})=t\,(w\cdot e_{i})=t\,(e_{i}\cdot w)=t\,w_{i},$$

which is (viii). By the last formula of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (ii), applied to the point $th$, and by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (v),

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

which is (ix).

Open sets. Let $U\subseteq\mathbb{R}^{n}$ be open and $a\in U$. Since $(\mathbb{R}^{n},d_{E})$ is a metric space by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence (i), Open Subset of a Metric Space (i) gives a real $r>0$ with $B_{d_{E}}(a,r)\subseteq U$. If $v\in\mathbb{R}^{n}$ and $\lVert v\rVert<r$, then $d_{E}(a,a+v)=\lVert v\rVert<r$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (ii), so $a+v\in B_{d_{E}}(a,r)$ by Open Ball in a Metric Space (ii), and hence $a+v\in U$. This proves (x). For (xi), take this $r$; if $|u|<r$, then $\lVert ue_{i}\rVert=|u|<r$ by (vii), so $a+ue_{i}\in U$.

Matrix rows. The entries $A_{kj}$, $j\in[n]$, are real numbers (Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (ii)), so the map $j\mapsto A_{kj}$ from $[n]$ to $\mathbb{R}$ is a point $\rho$ of $\mathbb{R}^{n}$ with $\rho_{j}=A_{kj}$, by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii). For any such $\rho$ and every $v\in\mathbb{R}^{n}$, Matrix-Vector Product (i) and The Dot Product on $\mathbb{R}^n$ (i) give

$$(Av)_{k}=\sum_{j=1}^{n}A_{kj}v_{j}=\sum_{j=1}^{n}\rho_{j}v_{j}=\rho\cdot v,$$

which is (xii).