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last edited 6 years ago by test1 |
Edit detail for FreeModule revision 3 of 8
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Editor: Bill Page
Time: 2011/03/10 16:05:35 GMT-8 |
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| Note: VectorSpace | ||
added:
From BillPage Thu Mar 10 16:05:26 -0800 2011
From: Bill Page
Date: Thu, 10 Mar 2011 16:05:26 -0800
Subject: VectorSpace
Message-ID: <20110310160526-0800@axiom-wiki.newsynthesis.org>
A [FreeModule] over a [Field] is a VectorSpace unfortunately this is not currently understood by Axiom:
\begin{axiom}
FreeModule(Fraction Integer,OrderedVariableList [e1,e1]) has VectorSpace(Fraction Integer)
\end{axiom}
Ref: http://en.wikipedia.org/wiki/Vector_space#Modules
A bi-module is a free module over a ring with generators indexed by an ordered 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 an ordered set and R is a ring (which may be non-commutative).
Ref: http://en.wikipedia.org/wiki/Free_module
axiom
)sh FreeModule
FreeModule(R: Ring,S: OrderedSet) is a domain constructor Abbreviation for FreeModule is FM This constructor is not exposed in this frame. ------------------------------- Operations -------------------------------- ?*? : (R, S) -> % ?*? : (S, R) -> % ?*? : (%, R) -> % ?*? : (R, %) -> % ?*? : (Integer, %) -> % ?*? : (PositiveInteger, %) -> % ?+? : (%, %) -> % ?-? : (%, %) -> % -? : % -> % ?=? : (%, %) -> Boolean 0 : () -> % coefficient : (%, S) -> R coefficients : % -> List(R) coerce : S -> % coerce : % -> OutputForm hash : % -> SingleInteger latex : % -> String leadingCoefficient : % -> R leadingMonomial : % -> S leadingSupport : % -> S map : ((R -> R), %) -> % monom : (S, R) -> % monomial : (R, S) -> % monomial? : % -> Boolean monomials : % -> List(%) reductum : % -> % retract : % -> S sample : () -> % support : % -> List(S) zero? : % -> Boolean ?~=? : (%, %) -> Boolean ?*? : (NonNegativeInteger, %) -> % construct : List(Record(k: S, c: R)) -> % constructOrdered : List(Record(k: S, c: R)) -> % leadingTerm : % -> Record(k: S, c: R) linearExtend : ((S -> R), %) -> R if R has COMRING listOfTerms : % -> List(Record(k: S, c: R)) numberOfMonomials : % -> NonNegativeInteger retractIfCan : % -> Union(S, "failed") subtractIfCan : (%, %) -> Union(%, "failed")
