Basis

The basis of a finite-dimensional algebra, as a list.

WikipediaBasis (linear algebra)MathWorldVectorBasisWikidataQ189569Encyclopedia of Mathematicsbasis

Basis(algebra)for a Clifford algebra, its

2^n

blades, ordered by grade then by generator:

\mathbb{H}

(1, i, j, k)

SageMathlist((algebra).basis())

Typed signatures

Type	Package	Applies
`untyped`	`hypercomplex`	always
`(any) -> unknown`	`structures`	always

Domain: Structures

Details

An algebra is an ordered list of generators; the families already carry the squares and the commutation rules
Constructors: CliffordAlgebra(p, q), MulticomplexAlgebra(n), SplitAlgebra(n), DualAlgebra(n), GrassmannAlgebra(n)
Named: Quaternions (also $\mathbb{H}$ ), BicomplexNumbers, TricomplexNumbers, SplitComplexNumbers, DualNumbers
No element constructor is needed — Dot threads a tuple of scalars over the basis

Examples

Basic #

$(1, i, j, k)$

\mathrm{Basis}(\mathrm{Quaternions})

Out

\bigl\lbrack1, f_1, f_2, f_1f_2\bigr\rbrack

the bicomplex basis

\mathrm{Basis}(\mathrm{MulticomplexAlgebra}(2))

Out

\bigl\lbrack1, i_1, i_2, i_1i_2\bigr\rbrack

Applications #

an element from a coefficient tuple

\mathrm{Dot}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \mathrm{Basis}(\mathrm{Quaternions}))

Out

4f_1f_2+2f_1+3f_2+1

About

Examples	3
Crosswalk	Wikipedia MathWorld Wikidata Encyclopedia of MathematicsSageMath

mappedsagepackages/oracle/src/mappings.ts