The command CanonicalBasis(S, T) will return a list W  of vectors, forms or tensors such that [i] span(S) = span(W) and [ii] the matrix whose rows are the coefficients of the elements of W with respect to T is in reduced row echelon form.

In the typical use of this command, S is a list of vectors or 1-forms on a manifold M and T is the standard basis for the tangent space or cotangent space of M.

The use of this command can dramatically simplify subsequent computations with the subspace spanned by S.

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

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

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

$\mathrm{S1}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}-\mathrm{D\_y}\,\mathrm{D\_x}+\mathrm{D\_y}\,\mathrm{D\_x}+\mathrm{D\_y}+\mathrm{D\_w}\right]\right)$

$\mathrm{S1}≔\left[\mathrm{D\_x}-\mathrm{D\_y}\,\mathrm{D\_x}+\mathrm{D\_y}\,\mathrm{D\_x}+\mathrm{D\_y}+\mathrm{D\_w}\right]$

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

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

$\mathrm{W1}≔\mathrm{CanonicalBasis}\left(\mathrm{S1}\,\mathrm{T1}\right)$

$\mathrm{W1}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_w}\right]$

$\mathrm{DGequal}\left(\mathrm{S1}\,\mathrm{W1}\right)$

$\mathrm{true}$

$\mathrm{S2}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}-\mathrm{D\_y}\,\mathrm{D\_x}+\mathrm{D\_y}\,\mathrm{D\_x}+\mathrm{D\_y}+\mathrm{D\_w}\right]\right)$

$\mathrm{S2}≔\left[\mathrm{D\_x}-\mathrm{D\_y}\,\mathrm{D\_x}+\mathrm{D\_y}\,\mathrm{D\_x}+\mathrm{D\_y}+\mathrm{D\_w}\right]$

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

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

$\mathrm{W2}≔\mathrm{CanonicalBasis}\left(\mathrm{S2}\,\mathrm{T2}\right)$

$\mathrm{W2}≔\left[\mathrm{D\_w}\,\mathrm{D\_y}\,\mathrm{D\_x}\right]$

$A≔\mathrm{Matrix}\left(\mathrm{GetComponents}\left(\mathrm{W2}\,\mathrm{T2}\right)\right)$

$A≔\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

$\mathrm{S3}≔\mathrm{evalDG}\left(\left[\mathrm{dx}\&w\mathrm{dy}+\mathrm{dy}\&w\mathrm{dz}-\mathrm{dy}\&w\mathrm{dw}\,\mathrm{dx}\&w\mathrm{dy}+\mathrm{dy}\&w\mathrm{dy}+\mathrm{dy}\&w\mathrm{dz}\,\mathrm{dy}\&w\mathrm{dz}+\mathrm{dy}\&w\mathrm{dw}\right]\right)$

$\mathrm{S3}≔\left[\mathrm{dx}\bigwedge \mathrm{dy}+\mathrm{dy}\bigwedge \mathrm{dz}-\mathrm{dy}\bigwedge \mathrm{dw}\,\mathrm{dx}\bigwedge \mathrm{dy}+\mathrm{dy}\bigwedge \mathrm{dz}\,\mathrm{dy}\bigwedge \mathrm{dz}+\mathrm{dy}\bigwedge \mathrm{dw}\right]$

$\mathrm{T3}≔\mathrm{GenerateForms}\left(\left[\left[\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}\,\mathrm{dw}\right]\right]\,\left[2\right]\right)$

$\mathrm{T3}≔\left[\mathrm{dx}\bigwedge \mathrm{dy}\,\mathrm{dx}\bigwedge \mathrm{dz}\,\mathrm{dx}\bigwedge \mathrm{dw}\,\mathrm{dy}\bigwedge \mathrm{dz}\,\mathrm{dy}\bigwedge \mathrm{dw}\,\mathrm{dz}\bigwedge \mathrm{dw}\right]$

$\mathrm{W3}≔\mathrm{CanonicalBasis}\left(\mathrm{S3}\,\mathrm{T3}\right)$

$\mathrm{W3}≔\left[\mathrm{dx}\bigwedge \mathrm{dy}\,\mathrm{dy}\bigwedge \mathrm{dz}\,\mathrm{dy}\bigwedge \mathrm{dw}\right]$

$\mathrm{DGequal}\left(\mathrm{S3}\,\mathrm{W3}\right)$

$\mathrm{true}$