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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.

(1)

(2)

(3)

Check orthogonality.

(4)

Matrix input, request nullspace only.

(5)

(6)

Check orthogonality.

(7)

For the same problem, request the nullspace as Matrix.

(8)

Check orthogonality.

(9)

See Also

LinearAlgebra/Details

LinearAlgebra[Modular]

LinearAlgebra[Modular][MatBasis]

LinearAlgebra[Modular][Mod]

LinearAlgebra[Modular][Multiply]

 


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