View source for Euclidean Spaces: Algebraic Structure and Inner Product

==Euclidean n-Space==
So far we have looked strictly at <math>\mathbb{R}</math> - the set of real numbers. We will now extend our reach to higher dimensions and looked at Euclidean <math>n</math>-space.
<blockquote style="background: white; border: 1px solid black; padding: 1em;"> 
:'''Definition:''' For each positive integer <math>n</math>, the '''Euclidean <math>n</math>-Space''' denoted <math>\mathbb{R}^n</math> is the set of all points <math>\mathbf{x} = (x_1, x_2, ..., x_n)</math> such that <math>x_1, x_2, ..., x_n \in \mathbb{R}</math>. The <math>k^{\mathrm{th}}</math> coordinate of the point <math>\mathbf{x}</math> is the real number <math>x_k</math>.
</blockquote>

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

The graphic in this [http://mathonline.wdfiles.com/local--files/euclidean-n-space/Screen%20Shot%202015-09-13%20at%209.25.03%20AM.png link] illustrates how we can visualize Euclidean <math>n</math>-space for <math>n = 1, 2, 3</math>.

Of course when <math>n \geq 4</math> it is practically impossibly to visualize Euclidean <math>n</math>-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 <math>\mathbf{x} = (x_1, x_2, ..., x_n)</math> for <math>x_1, x_2, ..., x_n \in \mathbb{R}</math> symbolically imply the existence of <math>n</math> mutually perpendicular axes that intersect at a point called the origin we denote by:

<div style="text-align: center;"><math>\begin{align} \quad \mathbf{0} = (0, 0, ..., 0) \end{align}</math></div>

The point <math>\mathbf{x}</math> is described to be located in respect to <math>\mathbf{0}</math>, i.e., the point <math>\mathbf{x}</math> is located <math>x_1</math> along the first axis, <math>x_2</math> along the the second axis, &#8230;, <math>x_n</math> along the <math>n^{\mathrm{th}}</math> axis. Sometimes we instead prefer to visualize <math>\mathbf{x}</math> as a vector (arrow) the starts at the origin and whose arrowhead ends at the point <math>(x_1, x_2, ..., x_n)</math>.

==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 <math>+</math> of addition and <math>\cdot</math> of multiplication of reals.

<blockquote style="background: white; border: 1px solid black; padding: 1em;"> 
:'''Definition:''' If <math>\mathbb{x} = (x_1, x_2, ..., x_n), \mathbb{y} = (y_1, y_2, ..., y_n) \in \mathbb{R}^n</math> then we define '''Equality''' <math>\mathbf{x} = \mathbf{y}</math> if and only if <math>x_k = y_k</math> for all <math>k \in \{ 1, 2, ..., n \}</math>.
</blockquote>

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

<blockquote style="background: white; border: 1px solid black; padding: 1em;"> 
: '''Definition:''' If <math>\mathbf{x} = (x_1, x_2, ..., x_n), \mathbf{y} = (y_1, y_2, ..., y_n) \in \mathbb{R}^n</math> then '''Addition''' is defined to be <math>\mathbf{x} + \mathbf{y} = (x_1 + y_1, x_2 + y_2, ..., x_n + y_n)</math> and <strong>Subtraction</strong> is defined to be <math>\mathbf{x} - \mathbf{y} = (x_1 - y_1, x_2 - y_2, ..., x_n - y_n)</math>.
</blockquote>

For example, consider the points <math>\mathbf{x} = (1, 4, 2, 6), \mathbf{y} = (3, -2, 0.5, \pi) \in \mathbb{R}^4</math>. Then:
<div style="text-align: center;"><math>\begin{align} \quad \mathbf{x} + \mathbf{y} = (1+3, 4 +(-2), 2 + 0.5, 6 + \pi) = (4, -2, 2.5, 6 + \pi) \end{align}</math></div>

And furthermore we have that:
<div style="text-align: center;"><math>\begin{align} \quad \mathbf{x} - \mathbf{y} = (1 - 3, 4 - (-2), 2 - 0.5, 6 - \pi) = (-2, 6, 1.5, 6 -\pi) \end{align}</math></div>

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

<blockquote style="background: white; border: 1px solid black; padding: 1em;"> 
:'''Definition:''' If <math>\mathbf{x} = (x_1, x_2, ..., x_n) \in \mathbb{R}^n</math> then '''Scalar Multiplication''' by the scalar <math>a \in \mathbb{R}</math> is defined to be 
:<math>a \mathbf{x} = a(x_1, x_2, ..., x_n) = (ax_1, ax_2, ..., ax_n)</math>.
</blockquote>

For example, consider the point <math>\mathbf{x} = (1, 2, 3, 4, 5) \in \mathbb{R}^5</math> and <math>a = 2 \in \mathbb{R}</math>. Then:
<div style="text-align: center;"><math>\begin{align} \quad a\mathbf{x} = 2(1, 2, 3, 4, 5) = (2, 4, 6, 8, 10) \end{align}</math></div>

==The Euclidean Inner Product==
<blockquote style="background: white; border: 1px solid black; padding: 1em;"> 
:'''Definition:''' Let <math>\mathbf{x} = (x_1, x_2, ..., x_n), \mathbf{y} = (y_1, y_2, ..., y_n) \in \mathbb{R}^n</math>. Then '''Euclidean Inner Product''' between <math>\mathbf{x}</math> and <math>\mathbf{y}</math> denoted <math>\mathbf{x} \cdot \mathbf{y}</math> is defined to be <math>\displaystyle{\mathbf{x} \cdot \mathbf{y} = x_1y_1 + x_2y_2 + ... + x_ny_n = \sum_{i=1}^{n} x_iy_i}</math>.
</blockquote>

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

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

<div style="text-align: center;"><math>\begin{align} \quad \mathbf{x} \cdot \mathbf{y} = (1 \cdot (-4), 4 \cdot (-3), 9 \cdot (-2), 16 \cdot (-1)) = (-4, -12, -18, -16) \end{align}</math></div>

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

<blockquote style="background: white; border: 1px solid black; padding: 1em;"> 
:'''Theorem 1:''' If <math>\mathbf{x} = (x_1, x_2, ..., x_n), \mathbf{y} = (y_1, y_2, ..., y_n)</math> and <math>a \in \mathbb{R}</math> then:
:'''a)''' <math>\mathbf{x} \cdot \mathbf{y} = \mathbf{y} \cdot \mathbf{x}</math>.
:'''b)''' <math>(a \mathbf{x}) \cdot \mathbf{y} = \mathbf{x} \cdot (a \mathbf{y})</math>.
</blockquote>

*'''Proof of a)''' Let <math>\mathbf{x}, \mathbf{y} \in \mathbb{R}^n</math>. Then:
<div style="text-align: center;"><math>\begin{align} \quad \mathbf{x} \cdot \mathbf{y} = x_1y_1 + x_2y_2 + ... + x_ny_n \end{align}</math></div>

*By the commutativity of multiplication, we have that:
<div style="text-align: center;"><math>\begin{align} \quad = y_1x_1 + y_2x_2 + ... + y_nx_n = \mathbf{y} \cdot \mathbf{x} \quad \blacksquare \end{align}</math></div>

*'''Proof of b)''' Let <math>\mathbf{x}, \mathbf{y} \in \mathbb{R}^n</math> and <math>a \in \mathbb{R}</math>. Then:
<div style="text-align: center;"><math>\begin{align} \quad (a\mathbf{x}) \cdot \mathbf{y}) = (ax_1, ax_2, ... ax_n) \cdot (y_1, y_2, ..., y_n) = ax_1y_1 + ax_2y_2 + ... + ax_ny_n \end{align}</math></div>

*Once again, by the commutativity of multiplication, we have that:
<div style="text-align: center;"><math>\begin{align} \quad = x_1(ay_1) + x_2(ay_2) + ... + x_n(ay_n) = \mathbf{x} \cdot (a \mathbf{y}) \quad \blacksquare \end{align}</math></div>


==Licensing==
Content obtained and/or adapted from:
* [http://mathonline.wikidot.com/euclidean-n-space Euclidean n-Space, mathonline.wikidot.com] under a CC BY-SA license
* [http://mathonline.wikidot.com/basic-operations-on-euclidean-n-space Basic Operations on Euclidean n-Space, mathonline.wikidot.com] under a CC BY-SA license
* [http://mathonline.wikidot.com/the-euclidean-inner-product The Euclidean Inner Product, mathonline.wikidot.com] under a CC BY-SA license