Linear combination synonym
The division ring F can be accessed as value of the attribute LeftActingDomain For a field F and a collection gens of vectors, VectorSpace returns the F -vector space spanned by the elements in gens. The optional argument zero can be used to specify the zero element of the space; zero must be given if gens is empty.
The optional linear combination synonym "basis" indicates that gens is known to be linearly independent over Fin particular the dimension of the vector space is immediately set; note that Basis SubspaceNC does the same as Subspaceexcept that it omits the check whether gens linear combination synonym a subset of V. The optional string "basis" indicates that gens is known to be linearly independent over F.
In this case the dimension of the subspace is immediately set, and both Subspace and SubspaceNC do not check whether gens really is linearly independent and whether gens is linear combination synonym subset of Linear combination synonym. Let F be a division ring and D a domain. If the elements in D form an F -vector space then AsVectorSpace returns this F -vector space, otherwise fail is returned.
AsVectorSpace can be used for example to view a given vector space as a vector space over a smaller or larger division ring. Let V be an F -vector space, and U a collection.
Otherwise fail is returned. For a vector space VTrivialSubspace linear combination synonym the subspace of V that consists of the zero vector in V. Called with a finite vector space vSubspaces returns the domain of all subspaces of V. Called with V and a nonnegative integer kSubspaces returns the domain of all k -dimensional subspaces of V. The domain of all subspaces of a finite vector space or of all subspaces of fixed dimension, as returned by Subspaces The operations described below make sense only for bases of finite dimensional linear combination synonym spaces.
In practice this means that the vector spaces must be low dimensional, that is, the dimension should not exceed a few hundred. Finally, of course there must be bases in GAP that really do the work. Basis Vvectors is either semi-echelonized or a relative basis. Other examples of bases of the third kind are canonical bases of finite fields and of linear combination synonym number fields. Examples are non-Gaussian row and matrix spaces, and subspaces of finite fields and abelian number fields that are themselves not fields.
In GAPa basis of a free left module is an object that knows how to compute coefficients w. Bases are constructed by Basis Each basis is an immutable list, the linear combination synonym -th entry being the i -th basis vector. Called with a free left F -module V as the only argument, Basis returns an F -basis of V whose vectors are not further specified.
If additionally a list vectors of vectors in V is given that linear combination synonym an F -basis of V then Basis returns this basis; if vectors is not linearly independent over F or does not generate V as a free left F -module then fail is returned.
BasisNC does the same as the two argument version of Basisexcept that it does not check whether vectors form a basis. If no basis vectors are prescribed then Basis need not compute basis vectors; in this case, the vectors are computed in the first call to BasisVectors If the vector space V supports a canonical basis then CanonicalBasis returns this basis, otherwise fail is returned.
The defining property linear combination synonym a canonical basis is that its vectors are uniquely determined by the vector space. The exact meaning of a canonical basis depends on the type of V. On the other linear combination synonym, one probably should install a Basis A relative basis is a basis of the free left module V that delegates the computation of coefficients etc. Let B be a basis of the free linear combination synonym module Vand vectors a list of vectors in V.
RelativeBasis checks whether vectors form a basis of Vand in this case a basis is returned in which vectors are the basis vectors; otherwise fail is returned.
For a vector space linear combination synonym B linear combination synonym, BasisVectors returns the list of basis vectors of B. The lists B and BasisVectors B are equal; linear combination synonym main purpose of BasisVectors is to provide access to a list of vectors that does not know linear combination synonym an underlying vector space. The reason why a basis stores a free left module is that otherwise one would have to store the basis vectors and the coefficient domain separately.
Let V be the underlying left module of the basis Band v a vector such that the family of v is the elements family of the family of V.
Then Coefficients Bv is the list of coefficients of v w. B if v lies in Vand fail otherwise. If B is a basis object see IsBasis It is useful to have a mutable basis of linear combination synonym free module when successively closures with new vectors are formed, since one does not want to create a new module and a corresponding basis for each step. So immutable bases and mutable bases are different categories of objects.
The only thing they have in common is that one can ask both for their basis vectors and for the coefficients of a given vector. Since Immutable produces an immutable copy of any GAP object, it would in principle be possible to construct linear combination synonym mutable basis that is in fact immutable. In the sequel, we will deal only with mutable bases that are in fact mutable GAP objects, hence these objects are unable to store attribute values.
Basic operations for immutable bases are NrBasisVectors Since mutable bases do not admit arbitrary changes of their lists of basis vectors, a mutable basis is not a list. Similar to the situation with bases cf. The generic method of MutableBasis returns a mutable basis that simply stores linear combination synonym immutable basis; clearly one wants to avoid this whenever possible with reasonable effort. There are mutable bases that store a mutable basis for a nicer module.
There are mutable bases that use special information to perform their tasks; examples are mutable bases of Gaussian row and matrix spaces. MutableBasis returns a mutable basis for the R -free module generated by the vectors in the list vectors.
The optional argument zero is the zero vector of the module; it must be given if linear combination synonym is empty. Note that vectors will in general not be the basis vectors of the mutable basis! Note that this operation is not an attribute, as it makes no sense to store the value. NrBasisVectors is used mainly as an equivalent of Dimension for the underlying left module in the case of immutable bases.
ImmutableBasis returns the immutable basis Bsay, with the same basis vectors as in the mutable basis MB. The second variant is used mainly for the case that one knows the module for the desired basis in advance, and if it has a nicer structure than the module known to MBfor example if it is an algebra. For a mutable basis MB over linear combination synonym coefficient ring Rsay, and a vector vIsContainedInSpan returns true is v lies in the R -span of the current basis vectors of MBand false linear combination synonym.
For a mutable basis MB over the coefficient ring Rsay, and a vector vCloseMutableBasis changes MB such that afterwards it describes the R -span of the former basis vectors together with v. Note that if v enlarges the dimension then this does in general not mean that v is simply added to the basis linear combination synonym of MB.
In this case, V is called a Gaussian vector space. Bases linear combination synonym Gaussian spaces can be computed using Gaussian elimination for a given list of vector space generators. If V is linear combination synonym row space then B is semi-echelonized if the matrix formed by its basis vectors has the property that the first nonzero element in each row is the identity of Fand all values exactly below these pivot elements are the zero of F cf.
If V is a matrix space then B is linear combination synonym if the matrix obtained by replacing each basis vector by the concatenation of its rows is semi-echelonized see above, cf.
If additionally a list vectors of vectors in V is given that forms a semi-echelonized basis of V then SemiEchelonBasis returns this basis; if vectors do not form a basis of V then fail is returned.
SemiEchelonBasisNC does the same as the two argument version of SemiEchelonBasisexcept that it is not checked whether vectors form a semi-echelonized basis. The result list can be used as action domain for the action of a matrix group via OnLines SiftedVector returns the residuum of v with respect to Bwhich is obtained by successively cleaning the pivot positions in v by linear combination synonym multiples of the basis vectors in B. So the result is the zero vector in V if and only if v lies in V.
GAP provides special functions to construct a particular linear mapping from images of given elements in the source, from a matrix of coefficients, or as a natural epimorphism. F -linear mappings with same source and same range can be added, so one can form vector spaces of linear mappings.
Let V and W be two left modules over the same left acting domain Rsay, and gens and imgs lists linear combination synonym the same linear combination synonym of elements in V and Wrespectively.
LeftModuleGeneralMappingByImages returns the general linear combination synonym with source V and range W that is defined by mapping the elements in gens to the corresponding elements linear combination synonym imgsand taking the R -linear closure.
LeftModuleHomomorphismByImages returns the left R -module homomorphism with source V and range W that is defined by mapping the elements in gens to the corresponding elements in imgs. If gens does not generate V or if the homomorphism does not exist i. For creating a possibly multi-valued mapping from V to W that linear combination synonym addition, multiplication, and scalar multiplication, LeftModuleGeneralMappingByImages The image of the i -th basis vector of BS is the linear combination of the basis vectors of BR with coefficients the i -th row of linear combination synonym.
A full hom module is a module of all R -linear mappings between two left R -modules. The function Hom A basis of a full hom module is called pseudo canonical basis if the matrices of its basis vectors w.
Let S and R be the source and the range, respectively, of each mapping in V. There are kinds of free R -modules for which efficient computations are possible because the elements are "nice", for example subspaces of full row modules or of full matrix modules. In other cases, a "nice" canonical basis is known that allows one to do the necessary computations in the corresponding row module, for example algebras given by structure constants.
In many other situations, one knows at least an isomorphism from the given linear combination synonym V to a "nicer" free left module Win the sense that for each vector in Vthe image in W can easily be computed, and analogously for each vector in Wone can compute the preimage in Linear combination synonym. This allows one to delegate computations w.
We call W the nice free left module of Vand C the nice basis of B. Note that it may happen that also C delegates questions to a "nicer" basis. The bijection between V and W is implemented by the functions NiceVector For a free left module V that is handled via linear combination synonym mechanism of nice bases, this attribute stores the associated free left module to which the tasks are delegated.
If v lies in the elements family of the family of V then NiceVector v is either fail or an element in the elements family of the family of W. Linear combination synonym r lies in the elements family of the family of W then UglyVector r is either fail or an element in the elements family of the family of V.
This allows one to implement for example a membership test for V using the membership test in W. For a free left module V that is handled via the mechanism of nice bases, this operation has to provide the necessary information if any for calls of NiceVector Let B be a basis of a free left module V that is handled via nice bases. Note that the result is fail if and only if the "basis vectors" stored in B are in fact not basis vectors.
In geometry a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planeswhile if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines.
This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, the objects which are hyperplanes may have different linear combination synonym. By its nature, it separates the space into two half spaces.
A hyperplane of an n -dimensional projective space does not have this property. If V is a vector space, one distinguishes "vector hyperplanes" which are linear subspacesand therefore must pass through the origin and "affine hyperplanes" which need not pass through the origin; they can be obtained linear combination synonym translation of a vector hyperplane.
A hyperplane in a Euclidean space separates that space into two half spacesand defines a reflection that fixes the hyperplane and interchanges those two half spaces. Several specific types of hyperplanes are defined with properties that are well suited for particular purposes. Some of these specializations are described here. An affine hyperplane is an affine subspace of codimension 1 in an affine space.
In the case of a real affine space, in other words when the coordinates are real numbers, this affine space separates the space into two half-spaces, which are the connected components of the complement of the hyperplane, and are given by the inequalities.
As an example, a point is a hyperplane in 1-dimensional space, a line is a hyperplane in 2-dimensional space, and a plane is a hyperplane in 3-dimensional space. A line in 3-dimensional space is not a hyperplane, and does not separate the space into two parts the complement of such a line is connected. Affine hyperplanes are used to linear combination synonym decision boundaries in many machine learning algorithms such as linear-combination oblique decision treesand perceptrons.
Such a hyperplane is the solution of a single linear equation. Linear combination synonym hyperplanesare used in projective geometry.
A projective subspace is a set of points with the property that for any two points of the set, all the points on the line determined by the two points are contained in the linear combination synonym. An affine hyperplane together with the associated points at infinity forms a projective hyperplane. One special case of a projective hyperplane is the infinite or ideal hyperplanewhich is defined with linear combination synonym set of all points at infinity.
In projective space, a hyperplane does not divide the space into two parts; rather, it takes two hyperplanes to separate points and divide up the space. The reason for this is that the space essentially "wraps around" so that both sides of a lone hyperplane linear combination synonym connected to each other.
The dihedral angle between two non-parallel hyperplanes of a Euclidean space is the angle between the corresponding normal vectors. The theory of polyhedrons and the dimension of the faces are analyzed by the looking at these intersections involving hyperplanes. From Wikipedia, the free encyclopedia. This article includes a list linear combination synonym referencesbut its sources remain unclear because it has insufficient inline citations.
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