Base (vector space)

In linear algebra , a basis is a subset of a vector space , with the help of which each vector of the space can be clearly represented as a finite linear combination . The coefficients of this linear combination are called the coordinates of the vector with respect to this base. One element of the basis is called a basis vector. If confusion with other basic terms (e.g. the shudder basis ) is to be feared, such a subset is also called the Hamel basis (after Georg Hamel ). A vector space generally has different bases, a change in the base forces a coordinate transformation . The Hamel base should not be confused with the base of a coordinate system , since these terms cannot be equated under certain conditions (e.g. curvilinear coordinates ).

Definition and basic terms

A basis of a vector space is a subset of with the following equivalent properties: ${\displaystyle V}$ ${\displaystyle B}$ ${\displaystyle V}$

1. Each element of can be represented as a linear combination of vectors and this representation is unique. ${\displaystyle V}$ ${\displaystyle B}$
2. ${\displaystyle B}$is a minimal generating system of , so every vector from can be represented as a linear combination of ( is a linear envelope of ) and this property no longer applies if an element from is removed. ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle B}$ ${\displaystyle V}$ ${\displaystyle B}$ ${\displaystyle B}$
3. ${\displaystyle B}$is a maximal linearly independent subset of . So if another element is added from to , the new set is no longer linearly independent. ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle B}$
4. ${\displaystyle B}$is a linearly independent generating system of . ${\displaystyle V}$

The elements of a basis are called basis vectors. If the vector space is a function space, the basis vectors are also called basis functions. A base can be described in the form using an index set , a finite base, for example, in the form . If such an index set is used, the family spelling is usually used to denote the base , i.e. H. instead . ${\displaystyle I}$ ${\displaystyle B=\{b_{i}:\;i\in I\}}$ ${\displaystyle B=\{b_{1},\dotsc ,b_{n}\}}$ ${\displaystyle I}$ ${\displaystyle b=(b_{i})_{i\in I}}$ ${\displaystyle B=\{b_{i}:\;i\in I\}}$

Note that in family notation, an order relation on the index set creates an arrangement of the basis vectors; then means “orderly basis”. This is used to describe the orientation of vector spaces. An index set with an order relation makes it possible to introduce orientation classes (handedness) under the bases. Examples: countable infinite base , finite base . ${\displaystyle I}$ ${\displaystyle b}$ ${\displaystyle b=(b_{i})_{i\in \mathbb {N} }}$ ${\displaystyle b=(b_{1},\dotsc ,b_{n})=(b_{i})_{i\in \{1,\dotsc ,n\}}}$

The coefficients that appear in the representation of a vector as a linear combination of vectors from the base are called the coordinates of the vector with respect to . These are elements of the body on which the vector space is based (e.g. or ). Together these form a coordinate vector , which, however, lies in another vector space, the coordinate space . Caution: Since the assignment of the coordinates to their respective base vectors is crucial, the base vectors themselves must be used for indexing in the absence of a common index set. ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle K}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle x=(x_{i})_{i\in B}}$ ${\displaystyle K^{B}}$

Although bases are usually written down as sets, an “indexing” given by an index set is therefore more practical. The coordinate vectors then have the form which is coordinate space . If there is an order relation, a sequence of coordinates is also created for the coordinate vector. In the example , the coordinate vector has the form ("numbering" of the coordinates). The coordinate space is here , for real or complex vector spaces so or . ${\displaystyle I}$ ${\displaystyle x=(x_{i})_{i\in I}}$ ${\displaystyle K^{I}}$ ${\displaystyle I}$ ${\displaystyle I=\{1,\dotsc ,n\}}$ ${\displaystyle x=(x_{1},\dotsc ,x_{n})}$ ${\displaystyle K^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}}$

Important properties

• Every vector space has a basis. Proof of this statement is given in the Proof of Existence section .
• All bases of a vector space contain the same number of elements. This number, which can also be an infinite cardinal number , is called the dimension of the vector space.
• A subset of a -Vektorraumes defining a linear mapping clearly , wherein the -th standard unit vector designated. ${\displaystyle \{b_{1},\dotsc ,b_{k}\}}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle K^{k}\to V,\quad e_{i}\mapsto b_{i}}$ ${\displaystyle e_{i}}$ ${\displaystyle i}$

This mapping is exactly then

• injective if they are linearly independent; ${\displaystyle b_{i}}$
• surjective , if they form a generating system; ${\displaystyle b_{i}}$
• bijective , if they form a basis. ${\displaystyle b_{i}}$

This characterization is carried over to the more general case of modules over rings , see Basis (module) .

e ₁ and e ₂ form a base of the plane.

Examples

• In the Euclidean plane there is what is known as the standard basis . In addition, two vectors form a basis in this plane if and only if they do not point in the same (or the opposite) direction. ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \{(1,0),(0,1)\}}$

• The standard basis of vector space is the set of canonical unit vectors . ${\displaystyle K^{n}}$ ${\displaystyle \{(1,0,\dotsc ,0),(0,1,0,\dotsc ,0),\dotsc ,(0,\dotsc ,0,1)\}}$

• The standard basis in the space of the matrices is formed by the standard matrices , in which there is exactly one entry and all other entries are. ${\displaystyle K^{m\times n}}$ ${\displaystyle 1}$ ${\displaystyle 0}$

• The zero vector space has dimension zero; its only basis is the empty set . ${\displaystyle \{0\}}$

• The base is mostly used as the vector space . A set is a basis of over if and only if is not a real number. As -vector space has a basis that cannot be specified explicitly. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \{1,i\}}$ ${\displaystyle \{a,b\}\subseteq \mathbb {C} \setminus \{0\}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\tfrac {a}{b}}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$

• The vector space of the polynomials over a body has the basis . But there are also many other bases that are more complicated to write down, but are more practical in specific applications, for example the Legendre polynomials . ${\displaystyle \{1,X,X^{2},X^{3},\dotsc \}}$

• In the vector space of the real number sequences the vectors form a linearly independent system, but no basis, because the sequence is not generated from it, for example , since a combination of infinitely many vectors is not a linear combination. ${\displaystyle \{(1,0,0,0,\dotsc ),(0,1,0,0,\dotsc ),(0,0,1,0,\dotsc ),\dotsc \}}$ ${\displaystyle (1,1,1,1,\dotsc )}$

Proof of the equivalence of the definitions

The following considerations outline a proof that the four characterizing properties, which are used in this article as the definition of the term base, are equivalent. (Zorn's axiom of choice or lemma is not required for this proof.)

• If every vector can be clearly represented as a linear combination of vectors in , then in particular is a generating system (by definition). If there is not a minimal generating system, then there is a real subset that is also a generating system. Now be an element of which is not in . It can then be represented in at least two different ways as a linear combination of vectors in , namely once as a linear combination of vectors in and once as . There is a contradiction and therefore it is minimal. So (1) → (2) holds. ${\displaystyle B}$ ${\displaystyle B}$
  ${\displaystyle B}$ ${\displaystyle B'}$ ${\displaystyle b^{*}}$ ${\displaystyle B}$ ${\displaystyle B'}$ ${\displaystyle b^{*}}$ ${\displaystyle B}$ ${\displaystyle B'}$ ${\displaystyle b^{*}=1\cdot b^{*}}$ ${\displaystyle B}$
  

• Every minimal generating system must be linearly independent. Because if is not linearly independent, then there is a vector in which can be represented as a linear combination of vectors in . But then every linear combination of vectors in can also be described by a linear combination of vectors in and would not be minimal. So (2) → (4) holds. ${\displaystyle B}$ ${\displaystyle b^{*}}$ ${\displaystyle B}$ ${\displaystyle B\setminus \{b^{*}\}}$ ${\displaystyle B}$ ${\displaystyle B\setminus \{b^{*}\}}$ ${\displaystyle B}$
  

• Every linearly independent generating system must be a maximally linearly independent set. If it were not maximally linearly independent, there would be one (which is not in ) which, together with linearly, would be independent. But can be represented as a linear combination of elements of what contradicts the linear independence. So (4) → (3) holds. ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle b^{*}}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle b^{*}}$ ${\displaystyle B}$
  

• A maximally linearly independent system is a generating system: Let be any vector. If in is included, then be of a linear combination of elements to write. But if is not contained in, then the set is a real superset of and therefore no longer linearly independent. The vectors that occur in a possible linear dependency cannot all be off , so one of them must (say ) be equal , with unequal 0. Therefore is . The uniqueness of this representation follows from the linear independence of . So (3) → (1) holds. ${\displaystyle B}$ ${\displaystyle b^{*}}$ ${\displaystyle b^{*}}$ ${\displaystyle B}$ ${\displaystyle b^{*}}$ ${\displaystyle B}$ ${\displaystyle b^{*}}$ ${\displaystyle B}$ ${\displaystyle B\ \cup \{b^{*}\}}$ ${\displaystyle B}$ ${\displaystyle b_{1},\dotsc ,b_{n}}$ ${\displaystyle a_{1}b_{1}+\dotsb +a_{n}b_{n}=0}$ ${\displaystyle B}$ ${\displaystyle b_{1}}$ ${\displaystyle b^{*}}$ ${\displaystyle a_{1}}$ ${\displaystyle b^{*}=-{\tfrac {1}{a_{1}}}(a_{2}b_{2}+\dotsb +a_{n}b_{n})}$ ${\displaystyle B}$
  

Proof of existence

With Zorn's lemma one can prove that every vector space must have a basis, even if one often cannot state it explicitly.

Let be a vector space. One would like to find a maximal linearly independent subset of the vector space. So it is obvious, the system of quantities ${\displaystyle V}$

${\displaystyle P:=\{X\subseteq V:\;X{\text{ linear unabhängig}}\}\,}$

to consider that is semi-ordered by the relation . One can now show: ${\displaystyle \subseteq }$

1. ${\displaystyle P}$is not empty (for example contains the empty set). If it not only consists of the zero vector , then every unit set with in and is also an element of . ${\displaystyle P}$ ${\displaystyle V}$ ${\displaystyle \{v\}}$ ${\displaystyle v}$ ${\displaystyle V}$ ${\displaystyle v\neq \mathbf {0} }$ ${\displaystyle P}$
2. For every chain is also in . ${\displaystyle C\subseteq P}$ ${\displaystyle \bigcup C=\bigcup _{X\in C}X=\{v:\exists X\in C:v\in X\}}$ ${\displaystyle P}$

From Zorn's lemma it follows that has a maximal element. The maximal elements of are now exactly the maximal linearly independent subsets of , i.e. the bases of . Therefore has a base, and it is also true that every linearly independent subset of is contained in a base of . ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle V}$

Basic supplementary set

Is a given set of linearly independent vectors and the above proof is based on ${\displaystyle T\subset V}$

${\displaystyle P:=\{X\subseteq V:\;T\subset X,\,X{\text{ linear unabhängig}}\}\,}$

off, one obtains the statement that a maximal element contains. Since such a maximal element turns out to be a basis of , it is shown that one can add any set of linearly independent vectors to a basis of . This statement is called the basic supplementary sentence. ${\displaystyle T}$ ${\displaystyle P}$ ${\displaystyle V}$ ${\displaystyle V}$

More statements about bases

• Exchange lemma from Steinitz (after E. Steinitz ): If one basis of a vector space and another vector different from the zero vector , one of the basis vectors can be "exchanged" for, ie. that is, there is an index so that is also a base of . This statement is often used to show that all bases in a vector space consist of the same number of vectors. ${\displaystyle v_{1},\dotsc ,v_{n}}$ ${\displaystyle V}$ ${\displaystyle w}$ ${\displaystyle V}$ ${\displaystyle w}$ ${\displaystyle 1\leq i\leq n}$ ${\displaystyle v_{1},\dotsc ,v_{i-1},w,v_{i+1},\dotsc ,v_{n}}$ ${\displaystyle V}$
  
• Each vector space is a free object above its base. This is a universal property of vector spaces in terms of category theory . Specifically, this means:

1. A linear mapping of one vector space into another vector space is already completely determined by the images of the basis vectors.
2. Any mapping of the base into the image space defines a linear mapping.

• In a -dimensional vector space over a finite field with elements there are ${\displaystyle d}$ ${\displaystyle q}$

${\displaystyle {\frac {1}{d!}}\prod _{k=0}^{d-1}\left(q^{d}-q^{k}\right)}$

different bases.

Basic concepts in special vector spaces

Real and complex vector spaces usually have additional topological structures. This structure can result in a basic term that deviates from the one described here.

Basis and dual basis in three-dimensional Euclidean vector space

In classical mechanics , the visual space is modeled with the three-dimensional Euclidean vector space (V³, ·) , which makes it particularly relevant. Euclidean vector spaces are u. a. defined by the fact that there is a scalar product "·" in them , which gives these vector spaces special and noteworthy properties.

In the three-dimensional Euclidean vector space there is exactly one dual basis for each basis , so that with the Kronecker delta δ the following applies: With an orthonormal basis , all basis vectors are normalized to length one and orthogonal in pairs . Then base and dual base match. ${\displaystyle {\vec {b}}_{1,2,3}}$ ${\displaystyle {\vec {b}}^{1,2,3}}$ ${\displaystyle {\vec {b}}_{i}\cdot {\vec {b}}^{j}=\delta _{i}^{j}.}$

Each vector can now be represented as a linear combination of the basic vectors: ${\displaystyle {\vec {v}}}$

${\displaystyle {\vec {v}}=\sum _{i=1}^{3}({\vec {v}}\cdot {\vec {b}}^{i}){\vec {b}}_{i}=\sum _{i=1}^{3}({\vec {v}}\cdot {\vec {b}}_{i}){\vec {b}}^{i}}$

This is because the difference vectors from to the vectors to the right of the equal sign are zero vectors . ${\displaystyle {\vec {v}}}$

The three-dimensional Euclidean vector space is a complete dot product space .

Hamel and Schauderbasis in scalar product spaces

When studying real or complex scalar product spaces , especially Hilbert spaces, there is another, more appropriate way of representing the elements of space. A basis consists of pairwise orthogonal unit vectors , and not only finite but also infinite sums (so-called series ) of basis vectors are permitted. Such a complete orthonormal system is never a basis in an infinitely dimensional space in the sense defined here; for a better distinction one also speaks of a shudder basis . The basic type described in this article is also called the Hamel base to distinguish it .

Auerbach bases

An Auerbach basis is a Hamel basis for a dense subspace in a normalized vector space , so that the distance of each basis vector from the product of the other vectors is equal to its norm .

Delimitation of the basic terms

• Both a Hamel base and a shudder base are linearly independent sets of vectors.
• A Hamel basis, or simply basis as described in this article, forms a generating system of vector space, i.e. That is, any vector of space can be represented as a linear combination of finitely many vectors of the Hamel base.
• In the case of a finite-dimensional real or complex scalar product space, an orthonormal basis (i.e. a minimal generating system of normalized, mutually perpendicular vectors) is both Hamel and Schauder basis.
• In the case of an infinite-dimensional, complete real or complex scalar product space (especially in an infinite-dimensional Hilbert space) a shudder basis is never a Hamel basis and vice versa. In the infinite-dimensional case, a Hamel basis often cannot even be orthonormalized.
• The Hamel basis of an infinitely dimensional, separable Hilbert space consists of an uncountable number of elements. A shudder base, however, in this case consists of a countable number of elements. There is therefore no Hilbert space of Hamel dimension . ${\displaystyle \aleph _{0}}$
• In Hilbert spaces, the basis (without addition) usually means a shudder basis , in vector spaces without a scalar product always a Hamel basis.

See also

• Base change (vector space)
• Standard base
• Complementary basis

literature

• Peter Knabner , Wolf Barth : Lineare Algebra. Basics and Applications. Springer Spectrum, Berlin / Heidelberg 2013, ISBN 978-3-642-32185-6 .
• Uwe Storch, Hartmut Wiebe: Textbook of Mathematics. Volume II: Linear Algebra. BI science, Mannheim a. a. 1990, ISBN 978-3-411-14101-2 .

Web links

Wikibooks: Linear Algebra: General Vector Spaces: Base and Dimension  - Learning and Teaching Materials

Individual evidence

1. ↑ In association theory , Steinitz's exchange lemma is placed within the framework of modular associations . It can be shown that it can be subsumed under Kurosch-Ore's theorem . In this context, one speaks of the exchange rate in modular associations . (See: Helmuth Gericke : Theory of Associations. , 2nd edition. Bibliographisches Institut, Mannheim 1967, pp. 143–146)