Important: The linalg package has been deprecated. Use the superseding command LinearAlgebra[Basis], instead.

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

If the argument V is a single vector, then this vector is returned as a set {V}.

If V is a set of vectors, then basis will return a basis for the vector space spanned by those vectors in terms of the original vectors.

If V is a matrix, then the routine computes a basis of the column space of V if the second optional argument is 'colspace', and it returns a basis of the row space of V if the second argument is 'rowspace'.

For an ordered basis, use a list of vectors rather than a set.

A basis for the zero-dimensional space is an empty set or list.

The command with(linalg,basis) allows the use of the abbreviated form of this command.

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

$\mathrm{v1}≔\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\:$

$\mathrm{v2}≔\left[\begin{array}{ccc}0& 1& 0\end{array}\right]\:$

$\mathrm{v3}≔\left[\begin{array}{ccc}0& 0& 1\end{array}\right]\:$

$\mathrm{v4}≔\left[\begin{array}{ccc}1& 1& 1\end{array}\right]\:$

$\mathrm{basis}\left(\left\{\mathrm{v1}\,\mathrm{v2}\,\mathrm{v3}\right\}\right)$

$\left\{\mathrm{v1}\,\mathrm{v2}\,\mathrm{v3}\right\}$

$\mathrm{basis}\left(\left[\mathrm{v3}\,\mathrm{v2}\,\mathrm{v1}\right]\right)$

$\left[\mathrm{v3}\,\mathrm{v2}\,\mathrm{v1}\right]$

$\mathrm{basis}\left(\left\{\mathrm{v1}\,\mathrm{v2}\,\mathrm{v3}\,\mathrm{v4}\right\}\right)$

$\left\{\mathrm{v1}\,\mathrm{v2}\,\mathrm{v3}\right\}$

$\mathrm{basis}\left(\left\{\left[\begin{array}{ccc}1& 1& 1\end{array}\right]\,\left[\begin{array}{ccc}2& 2& 2\end{array}\right]\,\left[\begin{array}{ccc}1& -1& 1\end{array}\right]\,\left[\begin{array}{ccc}2& -2& 2\end{array}\right]\,\left[\begin{array}{ccc}1& 0& 1\end{array}\right]\,\left[\begin{array}{ccc}0& 1& 1\end{array}\right]\right\}\right)$

$\left\{\left[\begin{array}{ccc}1& 1& 1\end{array}\right]\,\left[\begin{array}{ccc}2& \mathrm{-2}& 2\end{array}\right]\,\left[\begin{array}{ccc}0& 1& 1\end{array}\right]\right\}$

$A≔\mathrm{array}\left(\left[\left[1\,0\,0\right]\,\left[0\,1\,0\right]\,\left[0\,0\,1\right]\,\left[1\,1\,1\right]\right]\right)\:$

$\mathrm{basis}\left(A\,\'\mathrm{rowspace}\'\right)$

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