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	FriCAS Algebra
Complex Polynomial ←	↑	→ LinearOperator

FreeModule

Edit detail for FreeModule revision 8 of 8

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Editor: test1 Time: 2018/04/13 15:52:36 GMT+0
Note:

changed:
-A bi-module is a free module
-over a ring with generators indexed by an ordered set.
FreeModule implements free module
over a ring with generators indexed by a set.

changed:
-in the domain **R** where **S** is an ordered set
in the domain **R** where **S** is a set

FreeModule implements free module over a ring with generators indexed by a set. Each element can be expressed as a finite linear combination of generators. Only non-zero terms are stored.

This domain implements linear combinations of elements from the domain S with coefficients in the domain R where S is a set and R is a ring (which may be non-commutative).

Ref: http://en.wikipedia.org/wiki/Free_module

See: [PolySpad]?

A FreeModule over a [Field]? is a VectorSpace? unfortunately this is not currently understood by Axiom:

fricas

FreeModule(Fraction Integer,OrderedVariableList [e1,e1]) has VectorSpace(Fraction Integer)

$\label{eq1} \mbox{\rm false}$

(1)

Type: Boolean

Ref: http://en.wikipedia.org/wiki/Vector_space#Modules

Add:

        if R has Field then VectorSpace(R)
        ...
        if R has Field then
          if S has Finite then
            dimension():CardinalNumber == coerce size()$S
          else
             dimension():CardinalNumber == Aleph(0)

fricas

F2:=FreeModule(Fraction Integer,OrderedVariableList [e1,e1])

$\label{eq2}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e 1, e 1 ]))$

(2)

Type: Type

fricas

F2 has VectorSpace(Fraction Integer)

$\label{eq3} \mbox{\rm false}$

(3)

Type: Boolean

fricas

dimension()$F2

   The function dimension is not implemented in FreeModule(Fraction(
      Integer),OrderedVariableList([e1,e1])) .