Let S = [S_1, S_2, ..., S_k] be a list of  vectors, matrices, differential forms, or tensors. Then DGbasis(S) returns a sublist B = [S_i_1, S_i_2, ..., S_i_r] of S such that the elements of B define a basis for the subspace spanned by the elements of S.  Thus the elements of B are linearly independent and span(S) = span(B).

With the  keyword argument method = "real", the set S is viewed as a vector space of vectors, forms or tensors over the real numbers and the basis is calculated using real number coefficients instead of general expression (function) coefficients.

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

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

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

$\mathrm{S1}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}\,\mathrm{D\_x}+\mathrm{D\_y}\,\mathrm{D\_y}\,0\&mult\mathrm{D\_x}\,\mathrm{D\_y}-\mathrm{D\_z}\,\mathrm{D\_z}\,\mathrm{D\_x}+\mathrm{D\_y}+\mathrm{D\_w}\right]\right)$

$\mathrm{S1}≔\left[\mathrm{D\_x}\,\mathrm{D\_x}+\mathrm{D\_y}\,\mathrm{D\_y}\,0\mathrm{D\_x}\,\mathrm{D\_y}-\mathrm{D\_z}\,\mathrm{D\_z}\,\mathrm{D\_x}+\mathrm{D\_y}+\mathrm{D\_w}\right]$

$\mathrm{B1}≔\mathrm{DGbasis}\left(\mathrm{S1}\right)$

$\mathrm{B1}≔\left[\mathrm{D\_x}\,\mathrm{D\_x}+\mathrm{D\_y}\,\mathrm{D\_y}-\mathrm{D\_z}\,\mathrm{D\_x}+\mathrm{D\_y}+\mathrm{D\_w}\right]$

$\mathrm{GetComponents}\left(\mathrm{S1}\,\mathrm{B1}\right)$

$\left[\left[1\,0\,0\,0\right]\,\left[0\,1\,0\,0\right]\,\left[\mathrm{-1}\,1\,0\,0\right]\,\left[0\,0\,0\,0\right]\,\left[0\,0\,1\,0\right]\,\left[\mathrm{-1}\,1\,\mathrm{-1}\,0\right]\,\left[0\,0\,0\,1\right]\right]$

$\mathrm{Tools}:-\mathrm{CanonicalBasis}\left(\mathrm{B1}\,\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\,\mathrm{D\_w}\right]\right)$

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

$\mathrm{S2}≔\mathrm{evalDG}\left(\left[\mathrm{dx}\&t\mathrm{dx}\,\mathrm{dy}\&t\mathrm{dx}\,\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dx}\,\mathrm{dy}\&t\mathrm{dy}\right]\right)$

$\mathrm{S2}≔\left[\mathrm{dx}\mathrm{dx}\,\mathrm{dy}\mathrm{dx}\,\mathrm{dx}\mathrm{dx}+\mathrm{dy}\mathrm{dx}\,\mathrm{dy}\mathrm{dy}\right]$

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

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

$\mathrm{S3}≔\left[\mathrm{Matrix}\left(\left[\left[0\,1\right]\,\left[0\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[1\,0\right]\,\left[0\,1\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[1\,1\right]\,\left[0\,1\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[1\,0\right]\,\left[0\,-1\right]\right]\right)\right]$

$\mathrm{S3}≔\left[\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\,\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\,\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\,\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right]\right]$

$\mathrm{DGbasis}\left(\mathrm{S3}\right)$

$\left[\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\,\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\,\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right]\right]$

$\mathrm{S4}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}\,x\mathrm{D\_x}\,\left(3x-4\right)\mathrm{D\_x}\right]\right)$

$\mathrm{S4}≔\left[\mathrm{D\_x}\,x\mathrm{D\_x}\,\left(3x-4\right)\mathrm{D\_x}\right]$

$\mathrm{DGbasis}\left(\mathrm{S4}\right)$

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

$\mathrm{DGbasis}\left(\mathrm{S4}\,\mathrm{method}=''real''\right)$

$\left[\mathrm{D\_x}\,x\mathrm{D\_x}\right]$