help links subscribe changes refresh edit

	FriCAS Algebra
FreeModule ←	↑	→ OrderedVariableList

LinearOperator

last edited 2 years ago by Bill Page

Edit detail for LinearOperator revision 11 of 63

	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
Editor: page Time: 2011/04/09 01:34:24 GMT-7
Note: composition

changed:
-    id: ()->%
      ++ tensor product
    _/: (%,%) -> %
      ++ operator composition

changed:
-    T == CartesianTensor(1,dim,K)
-    Rep == Record(n:NonNegativeInteger, m:NonNegativeInteger, t:T)
    T ==> CartesianTensor(1,dim,K)
    NNI ==> NonNegativeInteger
    Rep == Record(domain:NNI, codomain:NNI, data:T)

changed:
-    arity(x:%):DirectProduct(2,NonNegativeInteger) == directProduct [rep(x).n,rep(x).m]
    dom(f:%):NNI == rep(f).domain
    cod(f:%):NNI == rep(f).codomain
    dat(f:%):T == rep(f).data

    arity(f:%):DirectProduct(2,NonNegativeInteger) == directProduct [dom f,cod f]

removed:
-    (x:% + y:%):% ==
-      rep(x).t=0 => per [rep(y).n,rep(y).m,rep(y).t]
-      rep(y).t=0 => per [rep(x).n,rep(x).m,rep(x).t]
-      rep(x).n ~= rep(y).n or rep(x).m ~= rep(y).m => error "arity"
-      per [rep(x).n,rep(x).m,rep(x).t+rep(y).t]
-
-    (x:% - y:%):% ==
-      rep(x).t=0 => per [rep(y).n,rep(y).m,-rep(y).t]
-      rep(y).t=0 => per [rep(x).n,rep(x).m,rep(x).t]
-      rep(x).n ~= rep(y).n or rep(x).m ~= rep(y).m => error "arity"
-      per [rep(x).n,rep(x).m,rep(x).t-rep(y).t]
-
-    1 == per [0,0,1]
-
-    --
-    -- f*g : A^n -> A^{m+p} = f:A^n -> A^m * g:A^n -> A^p
-    --
-    (x:% * y:%):% ==
-      output("* rep(x).n",rep(x).n::OutputForm)$OutputPackage
-      output("* rep(y).n",rep(y).n::OutputForm)$OutputPackage
-      rep(x).n ~= rep(y).n => error "arity"
-      r := product(rep(x).t, rep(y).t)
-      u := 1$DirectProduct(dim,K)::T
-      ud:= product(u,kroneckerDelta()$T)
-      for i in 1..rep(x).n repeat
-        --output("rank",rank(r)::OutputForm)$OutputPackage
-        --output("n",rep(x).n::OutputForm)$OutputPackage
-        r := contract(contract(ud,1,r,rep(x).n+1),1,rep(x).n+1)
-        --output("rank",rank(r)::OutputForm)$OutputPackage
-      per [rep(x).n,rep(x).m+rep(y).m,r]
-
-    -- repeated product
-    ((x:%)^(p:NonNegativeInteger)):% ==
-       q:=subtractIfCan(p,1)
-       q case NonNegativeInteger => x^q * x
-       per [rep(x).n,0,1]
-

changed:
-    (x:% + y:%):% ==
-      output("+ rep(x).m",rep(x).m::OutputForm)$OutputPackage
-      output("+ rep(y).m",rep(y).m::OutputForm)$OutputPackage
-      rep(x).m ~= rep(y).m  => error "arity"
-      output("+ rep(x).n",rep(x).n::OutputForm)$OutputPackage
-      output("+ rep(y).n",rep(y).n::OutputForm)$OutputPackage
-      r := product(rep(x).t, rep(y).t)
-      u := 1$DirectProduct(dim,K)::T
-      du:= product(kroneckerDelta()$T,u)
-      for i in 1..rep(y).m repeat
-        output("+ rank",rank(r)::OutputForm)$OutputPackage
-        output("+ rep(y).m",rep(y).m::OutputForm)$OutputPackage
-        r := contract(contract(du,1,r,rep(y).m+1),1,rep(y).m+2)
-        --output("rank",rank(r)::OutputForm)$OutputPackage
-      per [rep(x).n+rep(y).n,rep(y).m,r]
    (f:% + g:%):% ==
      dom(f) ~= dom(g) or cod(f) ~= cod(g) => error "arity"
      per [dom(f),cod(f),dat(f)+dat(g)]

    (f:% - g:%):% ==
      dom(f) ~= dom(f) or cod(g) ~= cod(g) => error "arity"
      per [dom(f),cod(f),dat(f)-dat(g)]

    --
    -- f/g : A^n -> A^p = f:A^n -> A^m / g:A^m -> A^p
    -- g*f : A^n -> A^p = g:A^m -> A^p * f:A^n -> A^m
    --
    (f:% / g:%):% ==
      cod(f) ~= dom(g) => error "arity"
      r:T := product(dat(f), dat(g))
      n:=dom(f)+1
      m:=n+cod(f)
      while m>n repeat
        r := contract(r,n,m)
        m:=m-1
      per [dom f,cod g,r]

    (g:% * f:%):% == f/g

    1:% == per [1,1,kroneckerDelta()$T]

    -- repeated composition
    (f:% ^ p:NNI):% ==
      cod(f) ~= dom(f) => error "arity"
      q:=subtractIfCan(p,1)
      q case NonNegativeInteger => f^q * f
      1

changed:
-    (x:% * p:NonNegativeInteger):% ==
    (f:% * p:NNI):% ==

changed:
-      q case NonNegativeInteger => x*q + x
-      per [0,rep(x).m,1]  -- need identity for + !!!
      q case NonNegativeInteger => f*q + f
      0

changed:
-      rep(x).n ~= rep(y).n or rep(x).m ~= rep(y).m => error "arity"
-      rep(x).t = rep(y).t
-
-    (x:K * y:%):% == per [rep(y).n,rep(y).m,x*rep(y).t]
-
-    (x:% * y:K):% == per [rep(x).n,rep(x).m,rep(x).t*y]
-
-    elt(x:%,y:%):% ==
-      r:=product(rep(x).t,rep(y).t)
-      yn:=rep(y).n  -- outputs of y
-      xm:=rep(x).m  -- inputs of x
-      while yn>0 and xm>0 repeat
-        output("yn",yn::OutputForm)$OutputPackage
-        output("xm",xm::OutputForm)$OutputPackage
-        output("rank",rank(r)::OutputForm)$OutputPackage
-        r:=contract(r,rep(y).n+xm,yn+rep(y).m)
-        yn:=subtractIfCan(yn,1)::NonNegativeInteger
-        xm:=subtractIfCan(xm,1)::NonNegativeInteger
-      per [rep(x).n+xm,yn+rep(y).m,r]
-
-    id():% == per [1,1,kroneckerDelta()$T]
      dom(x) ~= dom(y) or cod(x) ~= cod(y) => error "arity"
      dat(x) = dat(y)

    (x:K * y:%):% == per [dom(y),cod(y),x*dat(y)]

    (x:% * y:K):% == per [dom(x),cod(x),dat(x)*y]

changed:
-      #removeDuplicates([rep(y).n for y in x])~=1 or
-        #removeDuplicates([rep(y).m for y in x])~=1 => error "arity"
-      per [rep(first x).n+1,rep(first x).m,[rep(y).t for y in x]::T]$Rep
      #removeDuplicates([dom(y) for y in x])~=1 or
        #removeDuplicates([cod(y) for y in x])~=1 => error "arity"
      per [dom(first x)+1,cod(first x),[dat(y) for y in x]::T]$Rep

changed:
-      #removeDuplicates([rep(y).n for y in x])~=1 or
-        #removeDuplicates([rep(y).m for y in x])~=1 => error "arity"
-      per [rep(first x).n,rep(first x).m+1,[rep(y).t for y in x]::T]$Rep
-
-    coerce(x:%):OutputForm == (rep(x).t)::OutputForm
      #removeDuplicates([dom(y) for y in x])~=1 or
        #removeDuplicates([cod(y) for y in x])~=1 => error "arity"
      per [dom(first x),cod(first x)+1,[dat(y) for y in x]::T]$Rep

    coerce(x:%):OutputForm == (dat(x))::OutputForm

Linear transformations (operators) over n-dimensional cartesian vector spaces oveer a commutative ring . Members of this domain are morphisms $K^n \to K^m$ . Products, co-products and composition (grafting) of morphisms is implemented. Operators are represented internally as tensors.

Operator composition and products can be visualized by directed graphs (read from top to bottom) such as:

      n = 3     inputs

      \  |  /         \  /   /         \              |
       \ | /           \/   /           \            / \
        \|/             \  /           / \          /   \
        / \              \/           /   \         \   /
       /   \              \          /   / \         \ /
      /     \              \        /   /   \         |

      m = 2     outputs

Lines (edges) in the graph represent vectors, nodes represent operators. Horizontal juxtaposition represents product. Vertical juxtaposition represents composition.

spad

)abbrev domain LIN LinearOperator
LinearOperator(dim:NonNegativeInteger,K:CommutativeRing): Join(Ring,BiModule(K,K)) with
    arity: % -> DirectProduct(2,NonNegativeInteger)
    elt: (%,%) -> %
      ++ tensor product
    _/: (%,%) -> %
      ++ operator composition
    inp: List K -> %
      ++ incoming vector
    inp: List % -> %
    out: List K -> %
      ++ output vector
    out: List % -> %
    coerce: SquareMatrix(dim,K) -> %
    _*: (%,NonNegativeInteger) -> %
  == add
    import List NonNegativeInteger
    T ==> CartesianTensor(1,dim,K)
    NNI ==> NonNegativeInteger
    Rep == Record(domain:NNI, codomain:NNI, data:T)
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

    dom(f:%):NNI == rep(f).domain
    cod(f:%):NNI == rep(f).codomain
    dat(f:%):T == rep(f).data

    arity(f:%):DirectProduct(2,NonNegativeInteger) == directProduct [dom f,cod f]

    0 == per [0,0,0]

    --
    -- f+g : A^{n+m} -> A^p = f:A^n -> A^p + g:A^m -> A^p
    --
    (f:% + g:%):% ==
      dom(f) ~= dom(g) or cod(f) ~= cod(g) => error "arity"
      per [dom(f),cod(f),dat(f)+dat(g)]

    (f:% - g:%):% ==
      dom(f) ~= dom(f) or cod(g) ~= cod(g) => error "arity"
      per [dom(f),cod(f),dat(f)-dat(g)]

    --
    -- f/g : A^n -> A^p = f:A^n -> A^m / g:A^m -> A^p
    -- g*f : A^n -> A^p = g:A^m -> A^p * f:A^n -> A^m
    --
    (f:% / g:%):% ==
      cod(f) ~= dom(g) => error "arity"
      r:T := product(dat(f), dat(g))
      n:=dom(f)+1
      m:=n+cod(f)
      while m>n repeat
        r := contract(r,n,m)
        m:=m-1
      per [dom f,cod g,r]

    (g:% * f:%):% == f/g

    1:% == per [1,1,kroneckerDelta()$T]

    -- repeated composition
    (f:% ^ p:NNI):% ==
      cod(f) ~= dom(f) => error "arity"
      q:=subtractIfCan(p,1)
      q case NonNegativeInteger => f^q * f
      1

    -- repeated sum
    (f:% * p:NNI):% ==
      q:=subtractIfCan(p,1)
      q case NonNegativeInteger => f*q + f
      0

    (x:% = y:%):Boolean ==
      dom(x) ~= dom(y) or cod(x) ~= cod(y) => error "arity"
      dat(x) = dat(y)

    (x:K * y:%):% == per [dom(y),cod(y),x*dat(y)]

    (x:% * y:K):% == per [dom(x),cod(x),dat(x)*y]

    inp(x:List K):% == per [1,0,entries(x)::T]

    inp(x:List %):% ==
      #removeDuplicates([dom(y) for y in x])~=1 or
        #removeDuplicates([cod(y) for y in x])~=1 => error "arity"
      per [dom(first x)+1,cod(first x),[dat(y) for y in x]::T]$Rep

    out(x:List K):% == per [0,1,entries(x)::T]

    out(x:List %):% ==
      #removeDuplicates([dom(y) for y in x])~=1 or
        #removeDuplicates([cod(y) for y in x])~=1 => error "arity"
      per [dom(first x),cod(first x)+1,[dat(y) for y in x]::T]$Rep

    coerce(x:%):OutputForm == (dat(x))::OutputForm

spad

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/2505091042092146672-25px001.spad using 
      old system compiler.
   LIN abbreviates domain LinearOperator 
------------------------------------------------------------------------
   initializing NRLIB LIN for LinearOperator 
   compiling into NRLIB LIN 
   importing List NonNegativeInteger
   processing macro definition T$ ==> CartesianTensor(One,dim,K) 
   processing macro definition NNI ==> NonNegativeInteger 
   compiling local rep : $ -> Record(domain: NonNegativeInteger,codomain: NonNegativeInteger,data: CartesianTensor(One,dim,K))
      LIN;rep is replaced by x 
Time: 0.18 SEC.

   compiling local per : Record(domain: NonNegativeInteger,codomain: NonNegativeInteger,data: CartesianTensor(One,dim,K)) -> $
      LIN;per is replaced by x 
Time: 0.01 SEC.

   compiling local dom : $ -> NonNegativeInteger
Time: 0 SEC.

   compiling local cod : $ -> NonNegativeInteger
Time: 0 SEC.

   compiling local dat : $ -> CartesianTensor(One,dim,K)
Time: 0.02 SEC.

   compiling exported arity : $ -> DirectProduct(2,NonNegativeInteger)
Time: 0.05 SEC.

   compiling exported Zero : () -> $
Time: 0.01 SEC.

   compiling exported + : ($,$) -> $
Time: 0.02 SEC.

   compiling exported - : ($,$) -> $
Time: 0.02 SEC.

   compiling exported / : ($,$) -> $
Time: 0.02 SEC.

   compiling exported * : ($,$) -> $
Time: 0.01 SEC.

   compiling exported One : () -> $
Time: 0 SEC.

   compiling exported ^ : ($,NonNegativeInteger) -> $
Time: 0.02 SEC.

   compiling exported * : ($,NonNegativeInteger) -> $
Time: 0.02 SEC.

   compiling exported = : ($,$) -> Boolean
Time: 0.01 SEC.

   compiling exported * : (K,$) -> $
Time: 0.01 SEC.

   compiling exported * : ($,K) -> $
Time: 0 SEC.

   compiling exported inp : List K -> $
Time: 0.38 SEC.

   compiling exported inp : List $ -> $
Time: 0.10 SEC.

   compiling exported out : List K -> $
Time: 0 SEC.

   compiling exported out : List $ -> $
Time: 0.02 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0.01 SEC.

(time taken in buildFunctor:  10)

;;;     ***       |LinearOperator| REDEFINED

;;;     ***       |LinearOperator| REDEFINED
Time: 0.01 SEC.

   Cumulative Statistics for Constructor LinearOperator
      Time: 0.92 seconds

   finalizing NRLIB LIN 
   Processing LinearOperator for Browser database:
--->-->LinearOperator((arity ((DirectProduct 2 (NonNegativeInteger)) %))): Not documented!!!!
--------(elt (% % %))---------
--------(/ (% % %))---------
--------(inp (% (List K)))---------
--->-->LinearOperator((inp (% (List %)))): Not documented!!!!
--------(out (% (List K)))---------
--->-->LinearOperator((out (% (List %)))): Not documented!!!!
--->-->LinearOperator((coerce (% (SquareMatrix dim K)))): Not documented!!!!
--->-->LinearOperator((* (% % (NonNegativeInteger)))): Not documented!!!!
--->-->LinearOperator(constructor): Not documented!!!!
--->-->LinearOperator(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/LIN.NRLIB/LIN.lsp" (written 09 APR 2011 01:34:08 AM):

; /var/zope2/var/LatexWiki/LIN.NRLIB/LIN.fasl written
; compilation finished in 0:00:00.448
------------------------------------------------------------------------
   LinearOperator is now explicitly exposed in frame initial 
   LinearOperator will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/LIN.NRLIB/LIN

   >> System error:
   The bounding indices 163 and 162 are bad for a sequence of length 162.
See also:
  The ANSI Standard, Glossary entry for "bounding index designator"
  The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR

Construction operators: input and output

axiom

L:=LIN(2,FRAC POLY INT)

$\label{eq1}\hbox{\axiomType{LinearOperator}\ } (2, \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ })))$

(1)

Type: Type

axiom

A1:L:=inp[script(a,[[1,i]]) for i in 1..2]

$\label{eq2}\left[{a_{1, \: 1}}, \:{a_{1, \: 2}}\right]$

(2)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity A1

$\label{eq3}\left[ 1, \: 0 \right]$

(3)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

A2:L:=inp[script(a,[[2,i]]) for i in 1..2]

$\label{eq4}\left[{a_{2, \: 1}}, \:{a_{2, \: 2}}\right]$

(4)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

A:L:=inp[A1,A2]

$\label{eq5}\left[ \begin{array}{cc} {a_{1, \: 1}}&{a_{1, \: 2}} \ {a_{2, \: 1}}&{a_{2, \: 2}}$

(5)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity A

$\label{eq6}\left[ 2, \: 0 \right]$

(6)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

B1:L:=out[script(b,[[],[1,i]]) for i in 1..2]

$\label{eq7}\left[{b^{1, \: 1}}, \:{b^{1, \: 2}}\right]$

(7)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity B1

$\label{eq8}\left[ 0, \: 1 \right]$

(8)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

B2:L:=out[script(b,[[],[2,i]]) for i in 1..2]

$\label{eq9}\left[{b^{2, \: 1}}, \:{b^{2, \: 2}}\right]$

(9)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

B:L:=out[B1,B2]

$\label{eq10}\left[ \begin{array}{cc} {b^{1, \: 1}}&{b^{1, \: 2}} \ {b^{2, \: 1}}&{b^{2, \: 2}}$

(10)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity B

$\label{eq11}\left[ 0, \: 2 \right]$

(11)

Type: DirectProduct?(2,NonNegativeInteger?)

Products

$f<em>g : A^n \to A^{m+p} = f:A^n \to A^m </em> g:A^n \to A^p$

axiom

A12p := A1 * A2; A12p::OutputForm = A1::OutputForm * A2::OutputForm

   >> Error detected within library code:
   arity

Powers

axiom

A3p:=(A1*A1)*A1

   >> Error detected within library code:
   arity

axiom

B3p:=(B1*B1)*B1

   >> Error detected within library code:
   arity

Sums

$f+g : A^{n+m} \to A^p = f:A^n \to A^p + g:A^m \to A^p$

axiom

A12s := A1 + A2; A12s::OutputForm = A1::OutputForm + A2::OutputForm

$\label{eq12}{\left[{{a_{2, \: 1}}+{a_{1, \: 1}}}, \:{{a_{2, \: 2}}+{a_{1, \: 2}}}\right]}={{\left[{a_{1, \: 1}}, \:{a_{1, \: 2}}\right]}+{\left[{a_{2, \: 1}}, \:{a_{2, \: 2}}\right]}}$

(12)

Type: Equation(OutputForm?)

axiom

arity(A12s)::OutputForm = arity(A1)::OutputForm + arity(A2)::OutputForm

$\label{eq13}{\left[ 1, \: 0 \right]}={{\left[ 1, \: 0 \right]}+{\left[ 1, \: 0 \right]}}$

(13)

Type: Equation(OutputForm?)

axiom

B12s := B1 + B2; B12s::OutputForm = B1::OutputForm + B2::OutputForm

$\label{eq14}{\left[{{b^{2, \: 1}}+{b^{1, \: 1}}}, \:{{b^{2, \: 2}}+{b^{1, \: 2}}}\right]}={{\left[{b^{1, \: 1}}, \:{b^{1, \: 2}}\right]}+{\left[{b^{2, \: 1}}, \:{b^{2, \: 2}}\right]}}$

(14)

Type: Equation(OutputForm?)

axiom

arity(B12s)::OutputForm = arity(B1)::OutputForm + arity(B2)::OutputForm

$\label{eq15}{\left[ 0, \: 1 \right]}={{\left[ 0, \: 1 \right]}+{\left[ 0, \: 1 \right]}}$

(15)

Type: Equation(OutputForm?)

Multiplication

axiom

A3s:=(A1+A1)+A1

$\label{eq16}\left[{3 \ {a_{1, \: 1}}}, \:{3 \ {a_{1, \: 2}}}\right]$

(16)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(A3s)

$\label{eq17}\left[ 1, \: 0 \right]$

(17)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

test(A3s=A1+(A1+A1))

$\label{eq18} \mbox{\rm true}$

(18)

Type: Boolean

axiom

test(A3s=A1*3)

   >> Error detected within library code:
   arity

axiom

B3s:=(B1+B1)+B1

$\label{eq19}\left[{3 \ {b^{1, \: 1}}}, \:{3 \ {b^{1, \: 2}}}\right]$

(19)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(B3s)

$\label{eq20}\left[ 0, \: 1 \right]$

(20)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

test(B3s=B1+(B1+B1))

$\label{eq21} \mbox{\rm true}$

(21)

Type: Boolean

axiom

test(B3s=B1*3)

   >> Error detected within library code:
   arity

Expected error

axiom

B1A1:=B1*A1

$\label{eq22}\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}} \ {{b^{1, \: 1}}\ {a_{1, \: 2}}}&{{b^{1, \: 2}}\ {a_{1, \: 2}}}$

(22)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(B1A1)

$\label{eq23}\left[ 1, \: 1 \right]$

(23)

Type: DirectProduct?(2,NonNegativeInteger?)

Composition

axiom

AB11:=A1 B1

   Internal Error
   The function elt with signature hashcode is missing from domain 
      LinearOperator2(Fraction (Polynomial (Integer)))

Multiple inputs and outputs

axiom

W:L:=out[inp[inp([script(w,[[i,j],[k]]) for j in 1..2])$L for i in 1..2] for k in 1..2]

$\label{eq24}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {w_{1, \: 1}^{1}}&{w_{1, \: 2}^{1}} \ {w_{2, \: 1}^{1}}&{w_{2, \: 2}^{1}}$

(24)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

WB1:=W B1

   Internal Error
   The function elt with signature hashcode is missing from domain 
      LinearOperator2(Fraction (Polynomial (Integer)))

product and composition --Bill Page, Wed, 09 Mar 2011 18:53:36 -0800 reply

Another kind of diagram:

  ___ _       _        _  _
  ___ _)  =   _\_ _    _\/_
  _/          ___ _)   _/\_

Linear operators as morphisms --Bill Page, Thu, 10 Mar 2011 09:42:33 -0800 reply

If linear operators really are to be morphisms (in the sense of category theory) then they must have a domain and a co-domain that are vector spaces, not just an in-degree and out-degree. E.g.:

  Rep == Record(Dom:VectorSpace, Cod:VectorSpace, t:T)

But VectorSpace? is a category which would make Dom and Cod domains. The domains that currently satisfy VectorSpace? in Axiom are rather limited and seem oddly focused on number theory (finite fields). It is a good thing however that DirectProduct? is a conditional member of this cateogry.

axiom

DirectProduct(2,FRAC INT) has VectorSpace(FRAC INT)

$\label{eq25} \mbox{\rm true}$

(25)

Type: Boolean

Unfortunately:

axiom

DirectProduct(2,DirectProduct(2,FRAC INT)) has VectorSpace(FRAC INT)

$\label{eq26} \mbox{\rm false}$

(26)

Type: Boolean

The problem with VectorSpace? is that the domain [Vector]? is not a member of this category instead it satisfies VectoryCategory?.

Is it possible to treat tensors from CartesianTensor? as maps from VectorSpace? to VectorSpace?? We would like for example:

  T:DirectProduct(dim,DirectProduct(dim,FRAC INT)) -> DirectProduct(dim,FRAC INT))

To be a linear operator with two inputs and one output.

Modules --Bill Page, Thu, 10 Mar 2011 16:03:13 -0800 reply

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

axiom

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

$\label{eq27} \mbox{\rm true}$

(27)

Type: Boolean