This routine produce a generic linear combination of the elements in L. It can be used together with genKer or genIm to solve a system of homogeneous linear equations.
In this example we compute a generic and a particular linear combination of two particular differential 2-forms.
i1 : w = random newForm(2,1,2,"a")
2 2 2 2
o1 = (2x - 2x x + 6x - 3x x + 6x x )dx + (- 7x + 5x x - x - 2x x -
0 0 1 1 0 2 1 2 0 0 0 1 1 0 2
------------------------------------------------------------------------
2 2 2 2
x x + x )dx + (- 3x - 4x x + 7x + 2x x - 7x x - 3x )dx
1 2 2 1 0 0 1 1 0 2 1 2 2 2
o1 : DiffAlgForm
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i2 : h = random newForm(2,1,2,"a")
2 2 2 2 2
o2 = (x + 7x x + 3x + 2x x + x x + 8x )dx + (- x - 4x - 7x x - x x
0 0 1 1 0 2 1 2 2 0 0 1 0 2 1 2
------------------------------------------------------------------------
2 2 2 2
- x )dx + (3x - 4x x + 7x - 5x x + x )dx
2 1 0 0 1 1 0 2 2 2
o2 : DiffAlgForm
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i3 : linearComb({w,h},"a")
2 2
o3 = ((2a + a )x + (- 2a + 7a )x x + (6a + 3a )x + (- 3a + 2a )x x +
0 1 0 0 1 0 1 0 1 1 0 1 0 2
------------------------------------------------------------------------
2 2 2
(6a + a )x x + 8a x )dx + ((- 7a - a )x + 5a x x + (- a - 4a )x
0 1 1 2 1 2 0 0 1 0 0 0 1 0 1 1
------------------------------------------------------------------------
2
+ (- 2a - 7a )x x + (- a - a )x x + (a - a )x )dx + ((- 3a +
0 1 0 2 0 1 1 2 0 1 2 1 0
------------------------------------------------------------------------
2 2
3a )x + (- 4a - 4a )x x + (7a + 7a )x + (2a - 5a )x x - 7a x x +
1 0 0 1 0 1 0 1 1 0 1 0 2 0 1 2
------------------------------------------------------------------------
2
(- 3a + a )x )dx
0 1 2 2
o3 : DiffAlgForm
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i4 : random oo
2 2 2 2
o4 = (6x + 10x x + 18x - 2x x + 14x x + 16x )dx + (- 16x + 10x x -
0 0 1 1 0 2 1 2 2 0 0 0 1
------------------------------------------------------------------------
2 2 2
10x - 18x x - 4x x )dx + (- 16x x + 28x - 6x x - 14x x - 4x )dx
1 0 2 1 2 1 0 1 1 0 2 1 2 2 2
o4 : DiffAlgForm
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In this example we compute a generic differential 1-form that descends to the projective plane. Then, we impose another linear condition.
i5 : w = newForm(2,1,2,"a"); |
i6 : h = random newForm(2,2,1,"a"); |
i7 : L = genKer( (radial 2) _ w,w)
2 2 2
o7 = {x x dx - x dx , x x dx - x dx , x x dx - x x dx , x dx - x x dx ,
0 1 0 0 1 0 2 0 0 2 0 2 1 0 1 2 1 0 0 1 1
------------------------------------------------------------------------
2 2 2
x x dx - x x dx , x x dx - x dx , x dx - x x dx , x dx - x x dx }
1 2 0 0 1 2 1 2 1 1 2 2 0 0 2 2 2 1 1 2 2
o7 : List
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i8 : wr = linearComb(L,"a")
2 2 2
o8 = (a x x + a x + a x x + a x x + a x )dx + (- a x - a x x + a x x
0 0 1 3 1 1 0 2 4 1 2 6 2 0 0 0 3 0 1 2 0 2
------------------------------------------------------------------------
2 2 2
+ a x x + a x )dx + (- a x + (- a - a )x x - a x - a x x -
5 1 2 7 2 1 1 0 2 4 0 1 5 1 6 0 2
------------------------------------------------------------------------
a x x )dx
7 1 2 2
o8 : DiffAlgForm
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i9 : genKer(h ^ wr, wr)
5 2 5 3 3 2 5 3 2 2
o9 = {(-x + -x x + -x x + -x )dx + (- -x x + -x x + -x x + x )dx + (-
7 1 7 0 2 7 1 2 7 2 0 7 0 1 7 0 2 7 1 2 2 1
-----------------------------------------------------------------------
5 2 6 2 2 3
-x - -x x - -x - -x x - x x )dx
7 0 7 0 1 7 1 7 0 2 1 2 2
------------------------------------------------------------------------
}
-
o9 : List
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