Important: The linalg package has been deprecated. Use the superseding packages LinearAlgebra[RowSpace] and LinearAlgebra[ColumnSpace], instead.

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

The functions rowspace(A) and colspace(A) return a set of vectors that form a basis for the vector space spanned by the rows and columns of the matrix A, respectively.  The vectors are returned in a canonical form with leading entries 1.

The optional second parameter 'dim' is assigned the rank of A, which is the dimension of the row and column space.

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

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

$A≔\mathrm{matrix}\left(3\,2\,\left[2\,0\,3\,4\,0\,5\right]\right)$

$A≔\left[\begin{array}{cc}2& 0\\ 3& 4\\ 0& 5\end{array}\right]$

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

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

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

$\left\{\left[\begin{array}{ccc}1& 0& -\frac{15}{8}\end{array}\right]\,\left[\begin{array}{ccc}0& 1& \frac{5}{4}\end{array}\right]\right\}$

$B≔\mathrm{array}\left(\left[\left[0\,0\,0\,1\,0\right]\,\left[0\,x\,0\,0\,0\right]\,\left[0\,0\,y\,y\,0\right]\,\left[0\,x\,0\,1\,0\right]\right]\right)$

$B≔\left[\begin{array}{ccccc}0& 0& 0& 1& 0\\ 0& x& 0& 0& 0\\ 0& 0& y& y& 0\\ 0& x& 0& 1& 0\end{array}\right]$

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

$\left\{\left[\begin{array}{ccccc}0& 0& 1& 0& 0\end{array}\right]\,\left[\begin{array}{ccccc}0& 0& 0& 1& 0\end{array}\right]\,\left[\begin{array}{ccccc}0& 1& 0& 0& 0\end{array}\right]\right\}$

$C≔\mathrm{matrix}\left(3\,3\,\left[\frac{15}{4}\,-\frac{3}{4}\mathrm{sqrt}\left(10\right)\,\frac{1}{4}\mathrm{sqrt}\left(165\right)\,-\frac{3}{4}\mathrm{sqrt}\left(10\right)\,\frac{3}{2}\,-\frac{1}{4}\mathrm{sqrt}\left(66\right)\,\frac{1}{4}\mathrm{sqrt}\left(165\right)\,-\frac{1}{4}\mathrm{sqrt}\left(66\right)\,\frac{11}{4}\right]\right)$

$C≔\left[\begin{array}{ccc}\frac{15}{4}& -\frac{3\sqrt{10}}{4}& \frac{\sqrt{165}}{4}\\ -\frac{3\sqrt{10}}{4}& \frac{3}{2}& -\frac{\sqrt{66}}{4}\\ \frac{\sqrt{165}}{4}& -\frac{\sqrt{66}}{4}& \frac{11}{4}\end{array}\right]$

$\mathrm{colspace}\left(C\right)$

$\left\{\left[\begin{array}{ccc}0& 1& 0\end{array}\right]\,\left[\begin{array}{ccc}0& 0& 1\end{array}\right]\,\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\right\}$