Let S be a list of 1-forms and T a list of vectors.  Then Annihilator(S, T) calculates the subspace of vectors X in the span of T such that alpha(X) = 0 for all alpha in S.

Let S be a list of vectors and T a list of 1-forms.  Then Annihilator(S, T) calculates the subspace of 1-forms alpha in the span of T such that alpha(X) = 0 for all X in S.

If the optional argument T is not given, then T is taken to be the standard basis for the tangent space or cotangent space for the manifold M on which the elements of S are defined.

This command is part of the DifferentialGeometry package, and so can be used in the form Annihilator(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-Annihilator.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\,w\right]\,M\right)\:$

$\mathrm{S1}≔\left[\mathrm{dx}\,\mathrm{dy}\right]$

$\mathrm{S1}≔\left[\mathrm{dx}\,\mathrm{dy}\right]$

$\mathrm{T1}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

$\mathrm{T1}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

$\mathrm{T2}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\right]$

$\mathrm{T2}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\right]$

$\mathrm{Annihilator}\left(\mathrm{S1}\,\mathrm{T1}\right)$

$\left[\mathrm{D\_z}\right]$

$\mathrm{Annihilator}\left(\mathrm{S1}\,\mathrm{T2}\right)$

$\mathrm{Annihilator}\left(\mathrm{S1}\right)$

$\left[\mathrm{D\_w}\,\mathrm{D\_z}\right]$

$\mathrm{S2}≔\left[\mathrm{D\_y}\right]$

$\mathrm{S2}≔\left[\mathrm{D\_y}\right]$

$\mathrm{Annihilator}\left(\mathrm{S2}\right)$

$\left[\mathrm{dw}\,\mathrm{dz}\,\mathrm{dx}\right]$

$\mathrm{S3}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}-2\mathrm{D\_y}+\mathrm{D\_z}\,\mathrm{D\_y}+3\mathrm{D\_z}-\mathrm{D\_w}\right]\right)$

$\mathrm{S3}≔\left[\mathrm{D\_x}-2\mathrm{D\_y}+\mathrm{D\_z}\,\mathrm{D\_y}+3\mathrm{D\_z}-\mathrm{D\_w}\right]$

$\mathrm{A3}≔\mathrm{Annihilator}\left(\mathrm{S3}\right)$

$\mathrm{A3}≔\left[2\mathrm{dx}+\mathrm{dy}+\mathrm{dw}\,-7\mathrm{dx}-3\mathrm{dy}+\mathrm{dz}\right]$

$\mathrm{Matrix}\left(2\,2\,\left(i\,j\right)\mapsto \mathrm{Hook}\left(\mathrm{S3}\left[i\right]\,\mathrm{A3}\left[j\right]\right)\right)$

$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$