The bi-vector defined by vector fields $X$ and $Y$ is the rank 2, skew-symmetric, contravariant tensor field $T\=X\otimes Y-Y\otimes X$. More generally, the multi-vector defined by vector fields $X_1\,X_2\,..\.\,X_r$ is the rank $r$, skew-symmetric contravariant tensor field defined as the alternating sum of the tensor products of $X_1\,X_2\,..\.\,X_r$.

The vector fields $X_1\,X_2\,..\.\,X_r$ are linearly dependent if and only if the associated multi-vector vanishes.

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

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

$\mathrm{DGsetup}\left(\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]\,M\right)\:$

$\mathrm{X1}≔\mathrm{D\_x1}$

$\mathrm{X1}:=\mathrm{D\_x1}$

$\mathrm{X2}≔\mathrm{D\_x2}$

$\mathrm{X2}:=\mathrm{D\_x2}$

$\mathrm{MultiVector}\left(\left[\mathrm{X1}\,\mathrm{X2}\right]\right)$

$\mathrm{D\_x1}\mathrm{D\_x2}-\mathrm{D\_x2}\mathrm{D\_x1}$

$\mathrm{Y1}≔\mathrm{evalDG}\left(\mathrm{D\_x1}-\mathrm{D\_x2}\right)$

$\mathrm{Y1}:=\mathrm{D\_x1}-\mathrm{D\_x2}$

$\mathrm{Y2}≔\mathrm{evalDG}\left(\mathrm{D\_x2}-\mathrm{D\_x3}\right)$

$\mathrm{Y2}:=\mathrm{D\_x2}-\mathrm{D\_x3}$

$\mathrm{Y3}≔\mathrm{evalDG}\left(\mathrm{D\_x3}-\mathrm{D\_x4}\right)$

$\mathrm{Y3}:=\mathrm{D\_x3}-\mathrm{D\_x4}$

$\mathrm{MultiVector}\left(\left[\mathrm{Y1}\,\mathrm{Y2}\,\mathrm{Y3}\right]\right)$

$\mathrm{D\_x1}\mathrm{D\_x2}\mathrm{D\_x3}-\mathrm{D\_x1}\mathrm{D\_x2}\mathrm{D\_x4}-\mathrm{D\_x1}\mathrm{D\_x3}\mathrm{D\_x2}\+\mathrm{D\_x1}\mathrm{D\_x3}\mathrm{D\_x4}\+\mathrm{D\_x1}\mathrm{D\_x4}\mathrm{D\_x2}-\mathrm{D\_x1}\mathrm{D\_x4}\mathrm{D\_x3}-\mathrm{D\_x2}\mathrm{D\_x1}\mathrm{D\_x3}\+\mathrm{D\_x2}\mathrm{D\_x1}\mathrm{D\_x4}\+\mathrm{D\_x2}\mathrm{D\_x3}\mathrm{D\_x1}-\mathrm{D\_x2}\mathrm{D\_x3}\mathrm{D\_x4}-\mathrm{D\_x2}\mathrm{D\_x4}\mathrm{D\_x1}\+\mathrm{D\_x2}\mathrm{D\_x4}\mathrm{D\_x3}\+\mathrm{D\_x3}\mathrm{D\_x1}\mathrm{D\_x2}-\mathrm{D\_x3}\mathrm{D\_x1}\mathrm{D\_x4}-\mathrm{D\_x3}\mathrm{D\_x2}\mathrm{D\_x1}\+\mathrm{D\_x3}\mathrm{D\_x2}\mathrm{D\_x4}\+\mathrm{D\_x3}\mathrm{D\_x4}\mathrm{D\_x1}-\mathrm{D\_x3}\mathrm{D\_x4}\mathrm{D\_x2}-\mathrm{D\_x4}\mathrm{D\_x1}\mathrm{D\_x2}\+\mathrm{D\_x4}\mathrm{D\_x1}\mathrm{D\_x3}\+\mathrm{D\_x4}\mathrm{D\_x2}\mathrm{D\_x1}-\mathrm{D\_x4}\mathrm{D\_x2}\mathrm{D\_x3}-\mathrm{D\_x4}\mathrm{D\_x3}\mathrm{D\_x1}\+\mathrm{D\_x4}\mathrm{D\_x3}\mathrm{D\_x2}$

$Z≔\mathrm{evalDG}\left(a\mathrm{D\_x1}+b\mathrm{D\_x2}+c\mathrm{D\_x3}+d\mathrm{D\_x4}\right)$

$Z:=a\mathrm{D\_x1}\+b\mathrm{D\_x2}\+c\mathrm{D\_x3}\+d\mathrm{D\_x4}$

$T≔\mathrm{MultiVector}\left(\left[Z\,\mathrm{Y1}\,\mathrm{Y2}\,\mathrm{Y3}\right]\right)$

$T:=-\left(c\+d\+a\+b\right)\mathrm{D\_x1}\mathrm{D\_x2}\mathrm{D\_x3}\mathrm{D\_x4}\+\left(c\+d\+a\+b\right)\mathrm{D\_x1}\mathrm{D\_x2}\mathrm{D\_x4}\mathrm{D\_x3}\+\left(c\+d\+a\+b\right)\mathrm{D\_x1}\mathrm{D\_x3}\mathrm{D\_x2}\mathrm{D\_x4}-\left(c\+d\+a\+b\right)\mathrm{D\_x1}\mathrm{D\_x3}\mathrm{D\_x4}\mathrm{D\_x2}-\left(c\+d\+a\+b\right)\mathrm{D\_x1}\mathrm{D\_x4}\mathrm{D\_x2}\mathrm{D\_x3}\+\left(c\+d\+a\+b\right)\mathrm{D\_x1}\mathrm{D\_x4}\mathrm{D\_x3}\mathrm{D\_x2}\+\left(c\+d\+a\+b\right)\mathrm{D\_x2}\mathrm{D\_x1}\mathrm{D\_x3}\mathrm{D\_x4}-\left(c\+d\+a\+b\right)\mathrm{D\_x2}\mathrm{D\_x1}\mathrm{D\_x4}\mathrm{D\_x3}-\left(c\+d\+a\+b\right)\mathrm{D\_x2}\mathrm{D\_x3}\mathrm{D\_x1}\mathrm{D\_x4}\+\left(c\+d\+a\+b\right)\mathrm{D\_x2}\mathrm{D\_x3}\mathrm{D\_x4}\mathrm{D\_x1}\+\left(c\+d\+a\+b\right)\mathrm{D\_x2}\mathrm{D\_x4}\mathrm{D\_x1}\mathrm{D\_x3}-\left(c\+d\+a\+b\right)\mathrm{D\_x2}\mathrm{D\_x4}\mathrm{D\_x3}\mathrm{D\_x1}-\left(c\+d\+a\+b\right)\mathrm{D\_x3}\mathrm{D\_x1}\mathrm{D\_x2}\mathrm{D\_x4}\+\left(c\+d\+a\+b\right)\mathrm{D\_x3}\mathrm{D\_x1}\mathrm{D\_x4}\mathrm{D\_x2}\+\left(c\+d\+a\+b\right)\mathrm{D\_x3}\mathrm{D\_x2}\mathrm{D\_x1}\mathrm{D\_x4}-\left(c\+d\+a\+b\right)\mathrm{D\_x3}\mathrm{D\_x2}\mathrm{D\_x4}\mathrm{D\_x1}-\left(c\+d\+a\+b\right)\mathrm{D\_x3}\mathrm{D\_x4}\mathrm{D\_x1}\mathrm{D\_x2}\+\left(c\+d\+a\+b\right)\mathrm{D\_x3}\mathrm{D\_x4}\mathrm{D\_x2}\mathrm{D\_x1}\+\left(c\+d\+a\+b\right)\mathrm{D\_x4}\mathrm{D\_x1}\mathrm{D\_x2}\mathrm{D\_x3}-\left(c\+d\+a\+b\right)\mathrm{D\_x4}\mathrm{D\_x1}\mathrm{D\_x3}\mathrm{D\_x2}-\left(c\+d\+a\+b\right)\mathrm{D\_x4}\mathrm{D\_x2}\mathrm{D\_x1}\mathrm{D\_x3}\+\left(c\+d\+a\+b\right)\mathrm{D\_x4}\mathrm{D\_x2}\mathrm{D\_x3}\mathrm{D\_x1}\+\left(c\+d\+a\+b\right)\mathrm{D\_x4}\mathrm{D\_x3}\mathrm{D\_x1}\mathrm{D\_x2}-\left(c\+d\+a\+b\right)\mathrm{D\_x4}\mathrm{D\_x3}\mathrm{D\_x2}\mathrm{D\_x1}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(T\,''CoefficientSet''\right)$

$\left\{c\+d\+a\+b\,-d-c-b-a\right\}$