For all of the functions, the following statements hold:

The dimension and orientation of all Vectors (in all vector spaces) must be the same.

If constructor options are specified, each resulting Vector has the same specified options.

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

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

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

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

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

The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Vector constructor that builds the result. These options may also be provided in the form outputoptions=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions option, the latter takes precedence (regardless of the order).

These functions are part of the LinearAlgebra package, and so they can be used in the form Basis(..), SumBasis(..), or IntersectionBasis(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[Basis](..), LinearAlgebra[SumBasis](..), or LinearAlgebra[IntersectionBasis](..).

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

$\mathrm{v1}≔⟨1\left|0\right|0⟩\:$

$\mathrm{v2}≔⟨0\left|1\right|0⟩\:$

$\mathrm{v3}≔⟨0\left|0\right|1⟩\:$

$\mathrm{v4}≔⟨0\left|1\right|1⟩\:$

$\mathrm{v5}≔⟨1\left|1\right|1⟩\:$

$\mathrm{v6}≔⟨4\left|2\right|0⟩\:$

$\mathrm{v7}≔⟨3\left|0\right|-1⟩\:$

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

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

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

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

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

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

$\mathrm{Basis}\left(\mathrm{Vector}\left(4\,\mathrm{shape}=\mathrm{zero}\right)\right)$

$\varnothing$

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

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

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

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

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

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

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

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

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