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Student[LinearAlgebra]

 Basis
 return a basis for a vector space
 IntersectionBasis
 return a basis for the intersection of vector space(s)
 SumBasis
 return a basis for the direct sum of vector space(s)

 Calling Sequence Basis(V, options) IntersectionBasis(VS, options) SumBasis(VS, options)

Parameters

 V - Vector, list of Vectors, or set of Vectors VS - list whose elements represent vector spaces; each list element is a Vector or a list or set of Vectors whose span represents the vector space options - (optional) parameters; for a complete list, see LinearAlgebra[Basis]

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right):$
 > $\mathrm{v1}≔⟨1|0|0⟩:$
 > $\mathrm{v2}≔⟨0|1|0⟩:$
 > $\mathrm{v3}≔⟨0|0|1⟩:$
 > $\mathrm{v4}≔⟨0|1|1⟩:$
 > $\mathrm{v5}≔⟨1|1|1⟩:$
 > $\mathrm{v6}≔⟨4|2|0⟩:$
 > $\mathrm{v7}≔⟨3|0|-1⟩:$
 > $\mathrm{Basis}\left(\left[\mathrm{v1},\mathrm{v2},\mathrm{v2}\right]\right)$
 $\left[\left[\begin{array}{ccc}{1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{ccc}{0}& {1}& {0}\end{array}\right]\right]$ (1)
 > $\mathrm{Basis}\left(\left\{\mathrm{v4},\mathrm{v6},\mathrm{v7}\right\}\right)$
 $\left\{\left[\begin{array}{ccc}{0}& {1}& {1}\end{array}\right]{,}\left[\begin{array}{ccc}{4}& {2}& {0}\end{array}\right]{,}\left[\begin{array}{ccc}{3}& {0}& {-1}\end{array}\right]\right\}$ (2)
 > $\mathrm{Basis}\left(\mathrm{v1}\right)$
 $\left\{\left[\begin{array}{ccc}{1}& {0}& {0}\end{array}\right]\right\}$ (3)
 > $\mathrm{SumBasis}\left(\left[\left[\mathrm{v1},\mathrm{v2}\right],\left[\mathrm{v6},⟨0|1|0⟩\right]\right]\right)$
 $\left[\left[\begin{array}{ccc}{1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{ccc}{0}& {1}& {0}\end{array}\right]\right]$ (4)
 > $\mathrm{SumBasis}\left(\left[\left\{\mathrm{v1}\right\},\left[\mathrm{v2},\mathrm{v3}\right],\mathrm{v5}\right]\right)$
 $\left\{\left[\begin{array}{ccc}{1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{ccc}{0}& {1}& {0}\end{array}\right]{,}\left[\begin{array}{ccc}{0}& {0}& {1}\end{array}\right]\right\}$ (5)
 > $\mathrm{IntersectionBasis}\left(\left[\left[\mathrm{v1},\mathrm{v2},\mathrm{v3}\right],\left\{\mathrm{v4},\mathrm{v6},\mathrm{v7}\right\},\left[\mathrm{v3},\mathrm{v4},\mathrm{v5}\right]\right]\right)$
 $\left\{\left[\begin{array}{ccc}{1}& {1}& {1}\end{array}\right]{,}\left[\begin{array}{ccc}{0}& {1}& {1}\end{array}\right]{,}\left[\begin{array}{ccc}{0}& {0}& {1}\end{array}\right]\right\}$ (6)
 > $\mathrm{IntersectionBasis}\left(\left[\mathrm{v1},\left\{\mathrm{v3},\mathrm{v7}\right\}\right]\right)$
 $\left\{\left[\begin{array}{ccc}{3}& {0}& {0}\end{array}\right]\right\}$ (7)
 > $\mathrm{IntersectionBasis}\left(\left[\left[\mathrm{v1},\mathrm{v2}\right],\left[\mathrm{v3}\right]\right]\right)$
 $\left[\right]$ (8)