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 $(x_{1},x_{2},...,x_{n})$ .

Basic Operations Euclidean n-Space

In a moment we will look at some operations defined on Euclidean n-space that the reader should already be familiar with. Before we do though, the reader should note that all of the operations defined below are in compliance to the field axioms of the real numbers in that all of the operations below are all in conjunction with the operations $+$ of addition and $\cdot$ of multiplication of reals.

Definition: If $\mathbb {x} =(x_{1},x_{2},...,x_{n}),\mathbb {y} =(y_{1},y_{2},...,y_{n})\in \mathbb {R} ^{n}$ then we define Equality $\mathbf {x} =\mathbf {y}$ if and only if $x_{k}=y_{k}$ for all $k\in \{1,2,...,n\}$ .

For example, if $\mathbf {x} =(1,4,7)$ and $\mathbf {y} =(1,3,7)$ then $\mathbf {x} \neq \mathbf {y}$ since $4\neq 3$ .

Definition: If $\mathbf {x} =(x_{1},x_{2},...,x_{n}),\mathbf {y} =(y_{1},y_{2},...,y_{n})\in \mathbb {R} ^{n}$ then Addition is defined to be $\mathbf {x} +\mathbf {y} =(x_{1}+y_{1},x_{2}+y_{2},...,x_{n}+y_{n})$ and Subtraction is defined to be $\mathbf {x} -\mathbf {y} =(x_{1}-y_{1},x_{2}-y_{2},...,x_{n}-y_{n})$ .

For example, consider the points $\mathbf {x} =(1,4,2,6),\mathbf {y} =(3,-2,0.5,\pi )\in \mathbb {R} ^{4}$ . Then:

{\begin{aligned}\quad \mathbf {x} +\mathbf {y} =(1+3,4+(-2),2+0.5,6+\pi )=(4,-2,2.5,6+\pi )\end{aligned}}

And furthermore we have that:

{\begin{aligned}\quad \mathbf {x} -\mathbf {y} =(1-3,4-(-2),2-0.5,6-\pi )=(-2,6,1.5,6-\pi )\end{aligned}}

Note that in general $\mathbf {x} +\mathbf {y} \neq \mathbf {y} +\mathbf {x}$ which we are already familiar with in the case when $n=1$ .

Definition: If $\mathbf {x} =(x_{1},x_{2},...,x_{n})\in \mathbb {R} ^{n}$ then Scalar Multiplication by the scalar $a\in \mathbb {R}$ is defined to be

$a\mathbf {x} =a(x_{1},x_{2},...,x_{n})=(ax_{1},ax_{2},...,ax_{n})$ .

For example, consider the point $\mathbf {x} =(1,2,3,4,5)\in \mathbb {R} ^{5}$ and $a=2\in \mathbb {R}$ . Then:

{\begin{aligned}\quad a\mathbf {x} =2(1,2,3,4,5)=(2,4,6,8,10)\end{aligned}}

The Euclidean Inner Product

Definition: Let $\mathbf {x} =(x_{1},x_{2},...,x_{n}),\mathbf {y} =(y_{1},y_{2},...,y_{n})\in \mathbb {R} ^{n}$ . Then Euclidean Inner Product between $\mathbf {x}$ and $\mathbf {y}$ denoted $\mathbf {x} \cdot \mathbf {y}$ is defined to be $\displaystyle {\mathbf {x} \cdot \mathbf {y} =x_{1}y_{1}+x_{2}y_{2}+...+x_{n}y_{n}=\sum _{i=1}^{n}x_{i}y_{i}}$ .

Another term for the Euclidean inner product is simply "Dot Product".

Note that when $n=1$ that the Euclidean inner product is simply the operation $\cdot$ of multiplication. Let's look at an example for when $n>1$ . Consider the points $\mathbf {x} =(1,4,9,16),\mathbf {y} =(-4,-3,-2,-1)\in \mathbb {R} ^{4}$ . Then the dot product $\mathbf {x} \cdot \mathbf {y}$ is:

{\begin{aligned}\quad \mathbf {x} \cdot \mathbf {y} =(1\cdot (-4),4\cdot (-3),9\cdot (-2),16\cdot (-1))=(-4,-12,-18,-16)\end{aligned}}

We will now look at some nice properties of the Euclidean inner product that can be derived by the field axioms of $\mathbb {R}$ .

Theorem 1: If $\mathbf {x} =(x_{1},x_{2},...,x_{n}),\mathbf {y} =(y_{1},y_{2},...,y_{n})$ and $a\in \mathbb {R}$ then:

a) $\mathbf {x} \cdot \mathbf {y} =\mathbf {y} \cdot \mathbf {x}$ .

b) $(a\mathbf {x} )\cdot \mathbf {y} =\mathbf {x} \cdot (a\mathbf {y} )$ .