Vector spaces

Nebe, G.

doi:10.1107/97809553602060000541

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 1.5, pp. 29-30 | 1 | 2 |

Section 1.5.2.2. Vector spaces

Gabriele Nebe^a ^*

^a Abteilung Reine Mathematik, Universität Ulm, D-89069 Ulm, Germany
Correspondence e-mail: nebe@mathematik.uni-ulm.de

1.5.2.2. Vector spaces

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We shall now exploit the advantage of being independent of the dimensionality. The following definitions are independent of the dimension by replacing the specific dimensions 2 for the plane and 3 for the space by an unspecified integer number [n>0] . Although we cannot visualize four- or higher-dimensional objects, we can describe them in such a way that we are able to calculate with such objects and derive their properties.

Algebraically, an n-dimensional (real) vector v can be represented by a column of n real numbers. The n-dimensional real vector space $[{\bf V}_{n}]$ is then $[{\bf V}_{n} = \{ {\bi x} = \left(\matrix{ x_1 \cr \vdots \cr x_n } \right) \mid x_1,\ldots, x_n \in{\bb R} \}.]$ (In crystallography n is normally 3.) The entries $[x_1,\ldots, x_n]$ are called the coefficients of the vector $[{\bf x}]$ . On $[{\bf V}_{n}]$ one can naturally define an addition, where the coefficients of the sum of two vectors are the corresponding sums of the coefficients of the vectors. To multiply a vector by a real number, one just multiplies all its coefficients by this number. The null vector $[{\bf o} = \left(\matrix{ 0 \cr \vdots \cr 0 } \right) \in {\bf V}_{n}]$ can be distinguished, since $[{\bf v} + {\bf o} = {\bf v}]$ for all $[{\bf v }\in {\bf V}_{n}]$ .

The identification of a concrete vector space $[{\bf V}]$ with the vector space $[{\bf V}_{n}]$ can be done by choosing a basis of $[{\bf V}]$ . A basis of $[{\bf V}]$ is any tuple of n vectors $[{\bf B}: = ({\bf a}_{1},\ldots, {\bf a}_{n})]$ such that every vector of $[{\bf V}]$ can be written uniquely as a linear combination of the basis vectors: $[{\bf V} = \{ {\bf x} = x_{1}{\bf a}_{1} + \ldots + x_n {\bf a}_{n} \mid x_{1},\ldots, x_{n} \in {\bb R} \}]$ . Whereas a vector space has many different bases, the number n of vectors of a basis is uniquely determined and is called the dimension of $[{\bf V}]$ . The isomorphism (see Section 1.5.3.4 for a definition of isomorphism) $[\varphi _{\bf B}]$ between $[{\bf V}]$ and $[{\bf V}_{n}]$ maps the vector $[{\bf x} = x_{1}{\bf a}_{1} + \ldots + x_{n} {\bf a}_{n} \in {\bf V}]$ to its coefficient column $[{\bi x} = \left(\matrix{ x_{1} \cr \vdots \cr x_{n} } \right) \in {\bf V}_{n}]$ with respect to the chosen basis $[{\bf B}]$ . The mapping $[\varphi _{\bf B}]$ respects addition of vectors and multiplication of vectors with real numbers. Moreover, $[\varphi _{\bf B}]$ is a bijective mapping, which means that for any coefficient column $[{\bi x} \in {\bf V}_{n}]$ there is a unique vector $[{\bf x} \in {\bf V}]$ with $[\varphi _{\bf B}({\bf x}) = {\bi x}]$ . Therefore one can perform all calculations using the coefficient columns.

An important concept in mathematics is the automorphism group of an object. In general, if one has an object (here the vector space $[{\bf V}]$ ) together with a structure (here the addition of vectors and the multiplication of vectors with real numbers), its automorphism group is the set of all one-to-one mappings of the object onto itself that preserve the structure.

A bijective mapping $[\varphi: {\bf V} \rightarrow {\bf V} ]$ of the vector space $[{\bf V}]$ into itself satisfying $[\varphi ({\bf v} + {\bf w}) = \varphi ({\bf v}) +\varphi ({\bf w})]$ for all $[{\bf v},\,{\bf w} \in {\bf V}]$ and $[\varphi (x {\bf v}) =]$ $[x \varphi ({\bf v})]$ for all real numbers $[x\in {\bb R}]$ and all vectors $[{\bf v} \in {\bf V}]$ is called a linear mapping and the set of all these linear mappings is the linear group of $[{\bf V}]$ . To know the image of $[{\bf x} = x_{1} {\bf a}_{1} + \ldots + x_{n}{\bf a}_{n}]$ under a linear mapping $[\varphi]$ it suffices to know the images of the basis vectors $[{\bf a}_{1}, \ldots, {\bf a}_{n}]$ under $[\varphi]$ , since $[\varphi({\bf x}) =]$ $[x_{1} \varphi({\bf a}_{1}) + \ldots +]$ $[x_{n} \varphi({\bf a}_{n})]$ . Writing the coefficient columns of the images of the basis vectors as columns of a matrix $[{\bi A}]$ [i.e. $[\varphi({\bf a}_{i}) =]$ $[\sum _{j=1}^n {\bf a}_jA_{ji}]$ , $[i=1,\ldots, n]$ ], then the coefficient column of $[\varphi({\bf x})]$ with respect to the chosen basis $[{\bf B}]$ is just $[{\bi A} {\bi x}]$ . Note that the matrix of a linear mapping depends on the basis $[{\bf B}]$ of $[{\bf V}]$ . The matrix that corresponds to the composition of two linear mappings is the product of the two corresponding matrices. We have thus seen that the linear group of a vector space $[{\bf V}]$ of dimension n is isomorphic to the group of all invertible $[(n\times n)]$ matrices via the isomorphism $[\phi _{{\bf B}}]$ that associates to a linear mapping its corresponding matrix (with respect to the basis $[{\bf B}]$ ). This means that one can perform all calculations with linear mappings using matrix calculations.

In crystallography, the translation-vector space has an additional structure: one can measure lengths and angles between vectors. An n-dimensional real vector space with such an additional structure is called a Euclidean vector space, $[{\bf E}_{n}]$ . Its automorphism group is the set of all (bijective) linear mappings of $[{\bf E}_{n}]$ onto itself that preserve lengths and angles and is called the orthogonal group $[{\cal O}_{n}]$ of $[{\bf E}_n]$ . If one chooses the basis $[{\bf B} =]$ $[({\bf e}_1, \ldots, {\bf e}_{n})]$ to be the unit vectors (which are orthogonal vectors of length 1), then the isomorphism $[\phi _{\bf B}]$ above maps the orthogonal group $[{\cal O}_{n}]$ onto the set of all $[(n\times n)]$ matrices A with $[{\bi A}^{\rm T}{\bi A} =]$ $[{\bi I}]$ , the $[(n\times n)]$ unit matrix. ^T denotes the transposition operator, which maps columns to rows and rows to columns.

References

International Tables for Crystallography (2006). Vol. A1. ch. 1.5, pp. 29-30