Euclidean Spaces: Algebraic Structure and Inner Product

Euclidean n-Space

So far we have looked strictly at $\mathbb {R}$ - the set of real numbers. We will now extend our reach to higher dimensions and looked at Euclidean $n$ -space.

Definition: For each positive integer $n$ , the Euclidean $n$ -Space denoted $\mathbb {R} ^{n}$ is the set of all points $\mathbf {x} =(x_{1},x_{2},...,x_{n})$ such that $x_{1},x_{2},...,x_{n}\in \mathbb {R}$ . The $k^{\mathrm {th} }$ coordinate of the point $\mathbf {x}$ is the real number $x_{k}$ .

In the case where $n=1$ we have that Euclidean 1-space is simply the real line $\mathbb {R}$ . When $n=2$ we are looking at points $(x,y)$ in the plane, and when $n=3$ we are looking in at points $(x,y,z)$ in three-dimensional space.

The graphic in this link illustrates how we can visualize Euclidean $n$ -space for $n=1,2,3$ .

Of course when $n\geq 4$ it is practically impossibly to visualize Euclidean $n$ -space and so, we will usually talk merely about the points (or vectors) which make up the space. Like with the cases above, the point $\mathbf {x} =(x_{1},x_{2},...,x_{n})$ for $x_{1},x_{2},...,x_{n}\in \mathbb {R}$ symbolically imply the existence of $n$ mutually perpendicular axes that intersect at a point called the origin we denote by:

{\begin{aligned}\quad \mathbf {0} =(0,0,...,0)\end{aligned}}

The point $\mathbf {x}$ is described to be located in respect to $\mathbf {0}$ , i.e., the point $\mathbf {x}$ is located $x_{1}$ along the first axis, $x_{2}$ along the the second axis, …, $x_{n}$ along the $n^{\mathrm {th} }$ axis. Sometimes we instead prefer to visualize $\mathbf {x}$ as a vector (arrow) the starts at the origin and whose arrowhead ends at the point $(x_{1},x_{2},...,x_{n})$ .