Section 1.1 Euclidean Space
For every positive integer \(N\) we introduce the set
\begin{equation*}
\mathbb R^N
:=\{(x_1,x_2,\dots,x_N)\mid x_i\in\mathbb R, i=1,\dots,N\}
\end{equation*}
of arrays of real numbers of length \(N\text{.}\) For \(N=1\) we set \(\mathbb R^1:=\mathbb R\text{.}\) If \(N=2\) we can interpret \((x_1,x_2)\) as the coordinates of a point or the components of a vector in the plane as shown in Figure 1.1. Likewise for \(\mathbb R^3\) as shown in Figure 1.2 we can interpret \((x_1,x_2,x_3)\) as the coordinates of a point or the components of a vector in space. The space \(\mathbb R^N\) is called \(N\) dimensional Euclidean space.
In analogy to \(N=2\) or \(N=3\) we call the array
\begin{equation*}
\vect x:=(x_1,x_2,\dots,x_N)
\end{equation*}
an \(N\)-vector or simply a vector, and denote it by \(\vect x\text{.}\) The entries in such an array we call the components of a vector, or the coordinates of the point \(\vect x\in\mathbb R^N\text{.}\) We often write such a vector as a column vector
\begin{equation*}
\vect x=
\begin{bmatrix}
x_1\\ \vdots \\ x_N
\end{bmatrix}
\end{equation*}
as in case of vectors in the plane or in space. To save space we often write \((x_1,\dots,x_N)\) instead of a column vector.
Note that, even for \(N=2\) or \(N=3\text{,}\) the “coordinates” of a “vector” do not need to represent the coordinates of a point in the plane or in space. They can represent other quantities, indicating the dependence of one quantity upon others. Hence the number of “components” is not restricted to the physical “dimensions”. As an example consider the price of a car. It certainly depends on the price of the materials to manufacture it, the cost of labour, and the cost of shipping it to the dealer. Nobody would claim that these are the only cost factors, by limiting the length of an array to three! Try to think of other factors influencing the price of the car, give each a name, and you end up considering arrays of five, ten, or one hundred entries, which we then call components or coordinates.
In analogy to \(\mathbb R^2\) and \(\mathbb R^3\) we call \(N\) the dimension of \(\mathbb R^N\text{,}\) and call \(\mathbb R^N\) the \(N\)-dimensional Euclidean space, or an \(N\)-dimensional vector space. Hence we use the word “space” in two different meanings.
- The first is as “the unlimited three-dimensional expanse in which all material objects are located”, to quote the description given in the Free Dictionary 1 .
- The second meaning is the mathematical definition of the set of all arrays of real numbers of a certain length that also appears in the dictionary definition of “space”. Hence there is nothing mysterious about four or ten dimensional “spaces.” In linear algebra a precise mathematical definition of the “dimension” of a “vector space” is given. Intuitively, it represents the number of (independent) “degrees of freedom” of a system, which can be more than three! Even in physics “higher dimensional” problems appear.
As an example look at the equations describing the electric and magnetic fields. The Maxwell Equations show that they are not independent of each other, and a full description is only possible when looking at the two simultaneously. Hence it is convenient to look at one array of six components, the first three representing the electric, and the second three representing the magnetic field. The vector comprising the two has six components, and is usually called the “electro-magnetic field.” The field in general depends on at least four variables, namely the three spatial components and time. It may depend on more variables, so for instance the position and velocity of moving charged particles. Hence the “field” depends on \(3+1+3+3=10\) variables, and thus is a function defined on a subset of \(\mathbb R^{10}\) with values in \(\mathbb R^6\text{.}\)
www.thefreedictionary.com/space
