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# FreeOIModuleMap VectorInWidth -- apply a free OI-module map to an element

- Operator: SPACE
- Usage:
    `phi f`
- Inputs:
  - phi, an instance of the type FreeOIModuleMap,
  - f, an instance of the type VectorInWidth,
- Outputs:
  - an instance of the type VectorInWidth,

## Description

Evaluates phi at f.

    i1 : P = makePolynomialOIAlgebra(2, x, QQ);

    i2 : F = makeFreeOIModule(e, {1,2}, P);

    i3 : installGeneratorsInWidth(F, 3);

    i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);

    i5 : C = oiRes({b}, 2)

    o5 = 0: (e0, {3}, {-2})
         1: (e1, {5, 5}, {-4, -3})
         2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -4, -5, -4, -5, -5, -3, -2})

    o5 : OIResolution

    i6 : phi = C.dd_1

    o6 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})

    o6 : FreeOIModuleMap

    i7 : G = getBasisElements C_1

    o7 = {1e1                   , 1e1                   }
             5,{1, 2, 3, 4, 5},1     5,{1, 2, 3, 4, 5},2

    o7 : List

    i8 : phi G#0

    o8 = x   x   e0              - x   x   e0              - x   x   e0              + x   x   e0
          2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1

                                                                         10
    o8 : (QQ[x   , x   , x   , x   , x   , x   , x   , x   , x   , x   ])   in width 5, degrees {2, 2, 2, 2, 2, 2, 2, 2, 2, 2}
              2,5   2,4   2,3   2,2   2,1   1,5   1,4   1,3   1,2   1,1

    i9 : phi G#1

    o9 = -x   e0              + x   e0              + x   e0              - x   e0
           2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1

                                                                         10
    o9 : (QQ[x   , x   , x   , x   , x   , x   , x   , x   , x   , x   ])   in width 5, degrees {2, 2, 2, 2, 2, 2, 2, 2, 2, 2}
              2,5   2,4   2,3   2,2   2,1   1,5   1,4   1,3   1,2   1,1

## Ways to use this method:

- FreeOIModuleMap VectorInWidth -- apply a free OI-module map to an element

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OIGroebnerBases.m2:1887:0.