Important: The linalg package has been deprecated. Use the superseding packages, LinearAlgebra and VectorCalculus, instead.

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

The functions rowspan and colspan compute a spanning set for the row space and column space respectively of the matrix A. The matrix A must be a matrix of multivariate polynomials over the rationals.  The spanning set returned is a set of vectors (one-dimensional arrays) of polynomials.

If the optional second argument is given, it will be assigned the rank of A, which is the dimension of the row space and column space.

These functions use ``fraction-free'' Gaussian elimination to triangularize the matrix.  Hence, unlike linalg[rowspace] and linalg[colspace], no rational expressions are introduced during the elimination; as a result, if variables are later assigned particular values, division by zero cannot occur.

These functions are part of the linalg package, and so can be used in the form rowspan(..) only after performing the command with(linalg) or with(linalg, rowspan).  These functions can always be accessed in the long form linalg[rowspan](..).

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

$A≔\mathrm{matrix}\left(2\,2\,\left[a\,b\,c\,d\right]\right)$

$A≔\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$\mathrm{rowspan}\left(A\right)$

$\left\{\left[\begin{array}{cc}0& ad-bc\end{array}\right]\,\left[\begin{array}{cc}a& b\end{array}\right]\right\}$

$\mathrm{colspan}\left(A\right)$

$\left\{\left[\begin{array}{cc}a& c\end{array}\right]\,\left[\begin{array}{cc}0& ad-bc\end{array}\right]\right\}$

$B≔\mathrm{matrix}\left(3\,2\,\left[x+y\,y+2z\,x\,y\,y\,2z\right]\right)$

$B≔\left[\begin{array}{cc}x+y& y+2z\\ x& y\\ y& 2z\end{array}\right]$

$\mathrm{rowspan}\left(B\,'d'\right)$

$\left\{\left[\begin{array}{cc}x& y\end{array}\right]\,\left[\begin{array}{cc}0& 2xz-y^2\end{array}\right]\right\}$

$d$

$2$