The Basis(V) command returns a list or set of Vectors that forms a basis for the vector space spanned by the original Vectors, in terms of the original Vectors.  A basis for the 0-dimensional space is an empty list or set.

If V is a list of Vectors, the Basis(V) command returns a list of Vectors.  If V is a single Vector or a set of Vectors, a set of Vectors is returned.

The IntersectionBasis(VS) command returns a list or set of Vectors that forms a basis for the intersection of the vector spaces defined by the Vector in each list element of VS.  IntersectionBasis([]) returns the empty set.

If all the elements of VS are lists of Vectors, the IntersectionBasis(V) command returns a list of Vectors.  Otherwise, a set of Vectors is returned.

The SumBasis(VS) command returns a list or set of Vectors that forms a basis for the direct sum of the vector spaces defined by the Vector in each list element of VS.  SumBasis([]) returns the empty set.

If all the elements of VS are lists of Vectors, the SumBasis(V) command returns a list of Vectors.  Otherwise, a set of Vectors is returned.

All Vectors given in the V or VS parameters must have the same dimension and orientation.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$\mathrm{v1}≔⟨1\left|0\right|0⟩\:$

$\mathrm{v2}≔⟨0\left|1\right|0⟩\:$

$\mathrm{v3}≔⟨0\left|0\right|1⟩\:$

$\mathrm{v4}≔⟨0\left|1\right|1⟩\:$

$\mathrm{v5}≔⟨1\left|1\right|1⟩\:$

$\mathrm{v6}≔⟨4\left|2\right|0⟩\:$

$\mathrm{v7}≔⟨3\left|0\right|-1⟩\:$

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

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

$\mathrm{Basis}\left(\left\{\mathrm{v4}\,\mathrm{v6}\,\mathrm{v7}\right\}\right)$

$\left\{\left[\begin{array}{ccc}0& 1& 1\end{array}\right]\,\left[\begin{array}{ccc}4& 2& 0\end{array}\right]\,\left[\begin{array}{ccc}3& 0& \mathrm{-1}\end{array}\right]\right\}$

$\mathrm{Basis}\left(\mathrm{v1}\right)$

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

$\mathrm{SumBasis}\left(\left[\left[\mathrm{v1}\,\mathrm{v2}\right]\,\left[\mathrm{v6}\,⟨0\left|1\right|0⟩\right]\right]\right)$

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

$\mathrm{SumBasis}\left(\left[\left\{\mathrm{v1}\right\}\,\left[\mathrm{v2}\,\mathrm{v3}\right]\,\mathrm{v5}\right]\right)$

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

$\mathrm{IntersectionBasis}\left(\left[\left[\mathrm{v1}\,\mathrm{v2}\,\mathrm{v3}\right]\,\left\{\mathrm{v4}\,\mathrm{v6}\,\mathrm{v7}\right\}\,\left[\mathrm{v3}\,\mathrm{v4}\,\mathrm{v5}\right]\right]\right)$

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

$\mathrm{IntersectionBasis}\left(\left[\mathrm{v1}\,\left\{\mathrm{v3}\,\mathrm{v7}\right\}\right]\right)$

$\left\{\left[\begin{array}{ccc}3& 0& 0\end{array}\right]\right\}$

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