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Student[LinearAlgebra]

  

Basis

  

return a basis for a vector space

  

IntersectionBasis

  

return a basis for the intersection of vector space(s)

  

SumBasis

  

return a basis for the direct sum of vector space(s)

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Basis(V, options)

IntersectionBasis(VS, options)

SumBasis(VS, options)

Parameters

V

-

Vector, list of Vectors, or set of Vectors

VS

-

list whose elements represent vector spaces; each list element is a Vector or a list or set of Vectors whose span represents the vector space

options

-

(optional) parameters; for a complete list, see LinearAlgebra[Basis]

Description

• 

The Basis(V) command returns a list or set of Vectors that forms a basis for the vector space spanned by the original Vectors, in terms of the original Vectors.  A basis for the 0-dimensional space is an empty list or set.

  

If V is a list of Vectors, the Basis(V) command returns a list of Vectors.  If V is a single Vector or a set of Vectors, a set of Vectors is returned.

• 

The IntersectionBasis(VS) command returns a list or set of Vectors that forms a basis for the intersection of the vector spaces defined by the Vector in each list element of VS.  IntersectionBasis([]) returns the empty set.

  

If all the elements of VS are lists of Vectors, the IntersectionBasis(V) command returns a list of Vectors.  Otherwise, a set of Vectors is returned.

• 

The SumBasis(VS) command returns a list or set of Vectors that forms a basis for the direct sum of the vector spaces defined by the Vector in each list element of VS.  SumBasis([]) returns the empty set.

  

If all the elements of VS are lists of Vectors, the SumBasis(V) command returns a list of Vectors.  Otherwise, a set of Vectors is returned.

• 

All Vectors given in the V or VS parameters must have the same dimension and orientation.

Examples

withStudent[LinearAlgebra]:

v11|0|0:

v20|1|0:

v30|0|1:

v40|1|1:

v51|1|1:

v64|2|0:

v73|0|1:

Basisv1,v2,v2

100,010

(1)

Basisv4,v6,v7

301,011,420

(2)

Basisv1

100

(3)

SumBasisv1,v2,v6,0|1|0

100,010

(4)

SumBasisv1,v2,v3,v5

111,100,010

(5)

IntersectionBasisv1,v2,v3,v7,v4,v6,v3,v4,v5

111,011,001

(6)

IntersectionBasisv1,v3,v7

300

(7)

IntersectionBasisv1,v2,v3

(8)

See Also

LinearAlgebra[Basis]

Student[LinearAlgebra]

Student[LinearAlgebra][GramSchmidt]

Student[LinearAlgebra][LUDecomposition]

Student[LinearAlgebra][NullSpace]

Student[LinearAlgebra][RowSpace]

 


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