com.numericalmethod.suanshu.vector.doubles.operation

Class VectorSpace

java.lang.Object

com.numericalmethod.suanshu.vector.doubles.operation.VectorSpace

All Implemented Interfaces:

java.io.Serializable

public class VectorSpace
extends java.lang.Object
implements java.io.Serializable

A vector space is a set of vectors that are closed under some operations. The basis of a vector space is a set of vectors that, in a linear combination, can represent every vector in the space, and that no element of the set can be represented as a linear combination of the others. In other words, a basis is a linearly independent spanning set. The orthogonal complement A^⊥ of a subspace A of an inner product space V is the set of all vectors in V that are orthogonal to every vector in A. Informally, it is called the perp, short for perpendicular complement. For an m x n matrix A, where m ≥ n, the orthogonal basis are the orthogonalization of the columns. The orthogonal complement is A^⊥ = Null(A.t()).

See Also:

• Wikipedia: Basis (linear algebra)
• Wikipedia: Orthogonal complement

, Serialized Form

Constructor Summary

Constructors

Constructor and Description
VectorSpace(double epsilon, Vector... elements) Construct a vector space from an array of vectors.
VectorSpace(java.util.List<Vector> elements) Construct a vector space from a list of vectors.
VectorSpace(java.util.List<Vector> elements, double epsilon) Construct a vector space from a list of vectors.
VectorSpace(Matrix A) Construct a vector space from a matrix (a set of column vectors).
VectorSpace(Matrix A, double epsilon) Construct a vector space from a matrix (a set of column vectors).
VectorSpace(Vector... elements) Construct a vector space from an array of vectors.

Method Summary

Modifier and Type	Method and Description
java.util.List<Vector>	getBasis() Get the orthogonal basis.
java.util.List<Vector>	getComplement() Get the basis of the orthogonal complement.
Vector	getLinearSpan(double... d) Deprecated. Not supported yet.
Vector	getSpanningCoefficients(Vector b) Find a linear combination of the basis that best approximates a vector in the least square sense.
boolean	isInKernel(Vector b) Deprecated. Not supported yet.
boolean	isSpanned(Vector b) Check whether a vector is in the span of the basis.
int	rank() Get the rank of this vector space.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

VectorSpace

public VectorSpace(Matrix A,
                   double epsilon)

Construct a vector space from a matrix (a set of column vectors). This implementation computes the orthogonal basis and the orthogonal complement.

Parameters:

A - a matrix, i.e., (a set of column vectors)

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0. The ε is used to determine the numerical rank of the linear space.

VectorSpace

public VectorSpace(Matrix A)

Construct a vector space from a matrix (a set of column vectors). This implementation computes the orthogonal basis and the orthogonal complement.

Parameters:

A - a matrix, i.e., (a set of column vectors)

VectorSpace

public VectorSpace(java.util.List<Vector> elements,
                   double epsilon)

Construct a vector space from a list of vectors. This implementation computes the orthogonal basis and the orthogonal complement.

Parameters:

elements - a list of vectors

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0. The ε is used to determine the numerical rank of the linear space.

VectorSpace

public VectorSpace(java.util.List<Vector> elements)

Construct a vector space from a list of vectors. This implementation computes the orthogonal basis and the orthogonal complement.

Parameters:

elements - a list of vectors

VectorSpace

public VectorSpace(double epsilon,
                   Vector... elements)

Construct a vector space from an array of vectors. This implementation computes the orthogonal basis and the orthogonal complement.

Parameters:

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0. The ε is used to determine the numerical rank of the linear space.

elements - a list of vectors

VectorSpace

public VectorSpace(Vector... elements)

Construct a vector space from an array of vectors. This implementation computes the orthogonal basis and the orthogonal complement.

Parameters:

elements - a list of vectors

Method Detail

rank

public int rank()

Get the rank of this vector space.

Returns:

the rank

getBasis

public java.util.List<Vector> getBasis()

Get the orthogonal basis.

Returns:

the orthogonal basis

getComplement

public java.util.List<Vector> getComplement()

Get the basis of the orthogonal complement.

Returns:

the orthogonal basis of the nullspace

getLinearSpan

@Deprecated
public Vector getLinearSpan(double... d)

Deprecated. Not supported yet.

Get the linear span of the orthogonal basis from a set of coefficients.

Parameters:

d - coefficients

Returns:

a span of the basis

getSpanningCoefficients

public Vector getSpanningCoefficients(Vector b)

Find a linear combination of the basis that best approximates a vector in the least square sense. This implementation solves an OLS problem.

Parameters:

b - a vector

Returns:

the coefficients for a linear combination of basis; null if b is not in the vector space

See Also:

OLSSolver

isSpanned

public boolean isSpanned(Vector b)

Check whether a vector is in the span of the basis. That is, whether there exists a linear combination of the basis that equals the vector.

Parameters:

b - a vector

Returns:

true if b is in this vector space

isInKernel

@Deprecated
public boolean isInKernel(Vector b)

Deprecated. Not supported yet.

Check whether a vector is in the span of the the kernel/nullspace. That is, whether there exists a linear combination of the basis of the kernel that equals the vector.

Parameters:

b - a vector

Returns:

true if b is in the kernel/nullspace of this vector space