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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

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

   There are no library operations named VectorSpace 
      Use HyperDoc Browse or issue
                            )what op VectorSpace
      to learn if there is any operation containing " VectorSpace " in 
      its name.

   Cannot find a definition or applicable library operation named 
      VectorSpace with argument type(s) 
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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])

$\label{eq1}\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Fraction}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[ e 1, \: e 1 \right]}\right)}}\right)$

(1)

Type: Type

fricas

F2 has VectorSpace(Fraction Integer)

   There are no library operations named VectorSpace 
      Use HyperDoc Browse or issue
                            )what op VectorSpace
      to learn if there is any operation containing " VectorSpace " in 
      its name.

   Cannot find a definition or applicable library operation named 
      VectorSpace with argument type(s) 
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.