Euclidean N Space
Contents
Introduction
We can locate any point in a coordinate system by using pairs of numbers. In 2D geometry we need 2 numbers, and in 3D geometry we need 3 numbers to express a point. Most probably many people don't know beyond 3D how to express a point. If we want to express a point in 4 or 5 or higher dimensional space, what can we do? A quadruple of numbers \((2,4,3,1)\), for example, is used to represent a point in a 4 dimensional space, and the same goes for higher dimensions. Thus we can represent \(n\)-tuple of numbers in an \(n\)-dimensional space. Mathematically, there are many rules and properties of vector in these kind of space, which we'll discuss in this wiki. But we should keep in mind that it is impossible till now to geometrically visualize any tuple of numbers out of 3 dimensional space.
Definition
If \(n\) is a positive integer, then an ordered \(n\)-tuples is called a sequence of \(n\) real numbers and denoted by \(\left (a_{1},a_{2},a_{3},a_{4},\ldots,a_{n} \right )\). The set of all the ordered \(n\)-tuples is called the \(n\)-space and it is denoted by \(R^{n}\).
If the value of \(n\) is 2 or 3, we can say that they are ordered pair and ordered triple, respectively. We denote them by \(R^{2}\) and \(R^{3}\), which is generally called 2D space and 3D space, respectively.
Examples
If we want to represent a point on any space, suppose the space contains some liquid, and we want to represent a single point into the liquid. That point can be expressed by 3 numbers according to 3D space. But, if we consider the property of liquid, density is one of them. So, we must add an extra point for representing the point of liquid . That is \(d.\) Then the exact point of liquid into the whole liquid can be viewed as a set of \(4\)-tuples, and the form of representation should be \(v = (5,4,9,2)\), where \(d = 2\) describes the density of the liquid of that point.
Properties Of Vectors In \(R^{n}\)
If \(v=\left (v_{1},v_{2},v_{3}, \ldots, v_{n} \right )\) , \(u=\left (u_{1},u_{2},v_{3}, \ldots, u_{n} \right )\), \(w=\left (w_{1},w_{2},w_{3}, \ldots, w_{n} \right )\), and \(m\) and \(k\) are any scalar in \(R^{n},\) then
- \(u+v = v+u\)
- \(u+(v+w) = \left (u+v \right )+w\)
- \(u+0 = 0+u=0\)
- \(u+\left (-u \right ) = 0\)
- \( k(mu) = (km)u\)
- \(k(u+v) = ku+kv\)
- \((k+m)u = ku+mu\).
Inner Product Of Vectors In \(R^{n}\)
If \(v=\left (v_{1},v_{2},v_{3}, \ldots, v_{n} \right )\) and \(u=\left (u_{1},u_{2},v_{3}, \ldots, u_{n} \right )\) are 2 vectors in \(R^{n},\) then their inner product will be \[\langle u, v\rangle = u_{1}v_{1} + u_{2}v_{2}+\cdots +u_{n}v_{n} .\] For example, if \(u = (-1, 3, 5, 7)\) and \(v = (5, -4, 7, 0)\) in \(R^{4},\) then their inner product is \[\langle u, v\rangle = \langle -1, 5\rangle + \langle 3, -4\rangle +\langle 5, 7\rangle + \langle 7, 0\rangle = 18.\]
Norm And Distance Of Vectors In \(R^{n}\):
If \(u=\left (u_{1},u_{2},v_{3},\ldots,u_{n} \right )\) and \(v=\left (v_{1},v_{2},v_{3},\ldots,v_{n} \right )\) are 2 vectors in \(R^{n}\), then the norm of the vector \(u\) can be expressed by \(\left \| u \right \|=\sqrt{u_{1}^{2}+u_{2}^{2}+\cdots+u_{n}^{2}}.\) The distance between the two vectors \(u\) and \(v\) is \[\left \| u-v \right \|=\sqrt{\left ( u_{1}-v_{1} \right )^{2}+\left ( u_{2}-v_{2} \right )^{2}+\cdots+\left ( u_{n}-v_{n} \right )^{2}}.\]
Transformation Of Function From \(R^{n}\) To \(R\)
If a function \(f\) has the domain \(R^{n}\) and the function maps the co-domain to \(R\), then \(f\) is called the transformation or mapping from \(R^{n}\) to \(R\). It is denoted by \(f:R^{n}\rightarrow R\).
We can transform a function of \(R^{n}\) To \(R\) . It's very simple. See the example for understanding the transformation:
| Formula | Example | Classification | Description |
| \(f(x)\) | \(f(x)=x^2\) | Real-valued function of a real variable | Function from \(R\) to \(R\) |
| \(f(x,y)\) | \(f(x,y)=x^2+y^2\) | Real-valued function of two real variables | Function from \(R^2\) to \(R\) |
| \(f(x,y,z)\) | \(f(x,y,z)=x^2+y^2+z^2\) | Real-valued function of three real variables | Function from \(R^3\) to \(R\) |
| \(f(x_1,x_2,\ldots,x_n)\) | \(f(x_1,x_2,\ldots,x_n)=x_1^2+x_2^2+\cdots+x_n^2\) | Real-valued function of \(n\) real variables | Function from \(R^n\) to \(R\) |
Transformation Of Function From \(R^{n}\) To \(R^{m}\)
If a function \(f\) has the domain \(R^{n}\) and maps the co-domain to \(R^{m}\) , then \(f\) is called the transformation or mapping from \(R^{n}\) to \(R^{m}\). It is denoted by \(f:R^{n}\rightarrow R^{m}\). Here \(f:R^{n}\rightarrow R^{m}\) is called the operator on \(R^{n}\).
Suppose \(x_{1},x_{2},x_{3},\ldots,x_{n}\) are some real values and \(f\) is a function mapping these \(n\) values to \(m\) number of values \(w_{1},w_{2},w_{3},\ldots,w_{m}\). Then the transformation can be written by
\(w_{1}=f_{1}(x_{1},x_{2},x_{3},\ldots,x_{n})\)
\(w_{2}=f_{2}(x_{1},x_{2},x_{3},\ldots,x_{n})\)
\(w_{3}=f_{3}(x_{1},x_{2},x_{3},\ldots,x_{n})\)
\(\ldots\)
\(\ldots\)
\(w_{m}=f_{m}(x_{1},x_{2},x_{3},\ldots,x_{n})\).
The above \(m\) equations assign a unique point \(w_{1},w_{2},w_{3},\ldots, w_{m}\) in the space \(R^{m}\) to eact point \(x_{1},x_{2},x_{3},\ldots, x_{n}\) in \(R^{n}\). Actually, this is the transformation of function from \(R^{n}\) to \(R^{m}\).
A Transformation From \(R^{2}\) To \(R^{3}\)
Suppose that there are the following 3 equations:
\(w_{1}=x_{1}+x_{2}\)
\(w_{2}=3x_{1}x_{2}\)
\(w_{3}=x_{1}^{2}-x_{2}^{2}\).
Also, there is a function \(F\) that transforms a function of the space \(R^{2}\) to \(R^{3}\). It is denoted by \(F:R^{2}\rightarrow R^{3}\). By this transformation, the image of \((x_{1},x_{2})\) can be obtained by :
\[F\left ( x_{1},x_{2} \right ) = \left ( x_{1}+x_{2},\, \, 3x_{1}x_{2},\: \: x_{1}^{2}-x_{2}^{2} \right ).\]
You can use any value at the place of the variables.
Consider a Euclidean space \(\mathbb R^{n}\) and the 3 equations \[\begin{array} &y_{1}=x_{1}^{3}+x_{2}-1, &y_{2}=x_{1}^{3}, &y_{3}=x_{1}+x_{2}+2.\end{array}\] Now, a transformation \(T:\mathbb R^{2}\mapsto \mathbb R^{3}\) is defined by : \[T\left ( x_{1},x_{2} \right )=\left (x_{1}^{3}+x_{2}-1,x_{1}+x_{2}, x_{1}+x_{2}+2 \right ).\] Find \(T\left (2,8 \right )\).
Submit your answer as \(y_{1}+y_{2}-y_{3}\).