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LinearAlgebra[Modular]

  

Basis

  

compute Basis and Nullspace mod m

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Basis(m, A, orient, basis, nullsp)

Parameters

m

-

modulus

A

-

mod m Matrix or list or set of mod m Vectors

orient

-

orientation of Vectors in mod m Matrix

basis

-

output format of basis Vectors

nullsp

-

output format of nullspace Vectors

Description

• 

The Basis function computes a basis of the set of Vectors described by A, and returns that basis and the nullspace.

• 

The orient parameter is specified if A is a mod m Matrix.  It indicates to the Basis command whether the vectors of the input Matrix are in rows (orient=row) or columns (orient=column).

• 

The basis and nullsp parameters indicate the format in which the respective Vectors are returned by Basis. They can be any of the following values:

 

 

false

no output

row

list of mod m row vectors

column

list of mod m column vectors

Matrix[row]

mod m Matrix with vectors in the rows

Matrix[column]

mod m Matrix with vectors in the columns

 

 

• 

If neither basis or nullsp are false, both the basis and nullspace Vectors are returned as a two element sequence. If both are false, an error results.

  

For example, if Basis is called with basis=false and nullsp=column, the output would be a list of column Vectors for the nullspace.

• 

The Basis command works with a copy of the input data, using the MatBasis command to perform the computation, so if data storage or efficiency are a concern, the MatBasis command should be used.

• 

This command is part of the LinearAlgebra[Modular] package, so it can be used in the form Basis(..) only after executing the command with(LinearAlgebra[Modular]).  However, it can always be used in the form LinearAlgebra[Modular][Basis](..).

Examples

Three dimensional basis, two dimensional nullspace, and Vector input.

withLinearAlgebra[Modular]:

p2741

p:=2741

(1)

AseqModp,Vector[column]5,i→rand,float[8],i=1..3

A:=2543.1568.127.356.581.,430.1549.2376.1511.1839.,164.1946.211.49.2418.

(2)

B,NBasisp,A,column,row

B,N:=1.0.0.1972.1878.,0.1.0.1578.1735.,0.0.1.1724.2166.,769.1163.1017.1.0.,863.1006.575.0.1.

(3)

Check orthogonality.

seqseqMultiplyp,i,j,i=N,j=B

0.,0.,0.,0.,0.,0.

(4)

Matrix input, request nullspace only.

AModp,Matrix3,5,i,j→rand,integer[]

A:=30148075410494231980253314396720512635353265716172198

(5)

NBasisp,A,row,false,column

N:=12791956248110,327246470101

(6)

Check orthogonality.

seqMultiplyp,A,i,i=N

000,000

(7)

For the same problem, request the nullspace as Matrix.

NBasisp,A,row,false,Matrix[column]

N:=12793271956246424817011001

(8)

Check orthogonality.

Multiplyp,A,N

000000

(9)

See Also

LinearAlgebra/Details

LinearAlgebra[Modular]

LinearAlgebra[Modular][MatBasis]

LinearAlgebra[Modular][Mod]

LinearAlgebra[Modular][Multiply]

 


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