Given an homogeneous linear application in one variable, this routine returns a basis of the image of such linear application.
In this example we compute a basis of the image of the derivative of a projective differential 1-forms.
i1 : h = newForm(2,1,2,"a")
2 2 2 2
o1 = (a x + a x x + a x + a x x + a x x + a x )dx + (a x + a x x +
0 0 3 0 1 9 1 6 0 2 12 1 2 15 2 0 1 0 4 0 1
------------------------------------------------------------------------
2 2 2 2
a x + a x x + a x x + a x )dx + (a x + a x x + a x + a x x +
10 1 7 0 2 13 1 2 16 2 1 2 0 5 0 1 11 1 8 0 2
------------------------------------------------------------------------
2
a x x + a x )dx
14 1 2 17 2 2
o1 : DiffAlgForm
|
i2 : R = radial 2
o2 = x ax + x ax + x ax
0 0 1 1 2 2
o2 : DiffAlgField
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i3 : H = linearComb(genKer(R _ h, h),"a")
2 2 2
o3 = (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
o3 : DiffAlgForm
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i4 : genIm(diff H,H)
o4 = {-3x dx dx , -3x dx dx , x dx dx - x dx dx - 2x dx dx , -3x dx dx , -
0 0 1 0 0 2 2 0 1 1 0 2 0 1 2 1 0 1
------------------------------------------------------------------------
x dx dx - 2x dx dx - x dx dx , -3x dx dx , -3x dx dx , -3x dx dx }
2 0 1 1 0 2 0 1 2 1 1 2 2 0 2 2 1 2
o4 : List
|
It is possible to get a linear independent set of elements using this routine:
i5 : w1=random newForm(2,1,2,"a"); |
i6 : w2=random newForm(2,1,2,"a"); |
i7 : w3=w1+w2; |
i8 : u=linearComb({w1,w2,w3},"a");
|
i9 : genIm(u,u)
2 2 2 2
o9 = {(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
------------------------------------------------------------------------
2 2 2 2 2
, (x + 7x x + 3x + 2x x + x x + 8x )dx + (- x - 4x - 7x x - x
0 0 1 1 0 2 1 2 2 0 0 1 0 2 1
- ---------------------------------------------------------------------
2 2 2 2
- x )dx + (3x - 4x x + 7x - 5x x + x )dx
2 1 0 0 1 1 0 2 2 2
------------------------------------------------------------------------
x }
2
---
o9 : List
|
i10 : #oo o10 = 2 |