Editing Multilinear algebra (section)

==Origin==
While many theoretical concepts and applications involve [[Vector space|single vectors]], mathematicians such as [[Hermann Grassmann]] considered structures involving pairs, triplets, and [[multivector]]s that generalize [[Vector (mathematics and physics)|vectors]]. With multiple combinational possibilities, the space of [[multivector]]s expands to 2<sup>''n''</sup> dimensions, where ''n'' is the dimension of the relevant vector space.<ref>{{cite book |last=Grassmann |first=Hermann |url={{GBurl|yeGPeaPVLKoC|pg=PP1}} |title=Extension Theory |publisher=[[American Mathematical Society]] |year=2000 |isbn=978-0-8218-9049-3 |translator-last=Kannenberg |translator-first=Lloyd |trans-title=Die Ausdehnungslehre |orig-year=1862}}</ref> The [[Determinant#Abstract formulation|determinant can be formulated abstractly]] using the structures of multilinear algebra. 

Multilinear algebra appears in the study of the mechanical response of materials to stress and strain, involving various moduli of [[Elasticity (physics)|elasticity]]. The term "[[tensor]]" describes elements within the multilinear space due to its added structure. Despite Grassmann's early work in 1844 with his ''[[Ausdehnungslehre]]'', which was also republished in 1862, the subject was initially not widely understood, as even ordinary linear algebra posed many challenges at the time.

The concepts of multilinear algebra find applications in certain studies of [[multivariate calculus]] and [[Manifold|manifolds]], particularly concerning the [[Jacobian matrix]]. [[Differential (infinitesimal)|Infinitesimal differentials]] encountered in single-variable calculus are transformed into [[differential forms]] in [[Multivariable calculus|multivariate calculus]], and their manipulation is carried out using [[exterior algebra]].<ref>{{cite book |last=Fleming |first=Wendell H. |title=Functions of Several Variables |date=1977 |publisher=Springer |isbn=978-1-4684-9461-7 |edition=2nd |series=Undergraduate Texts in Mathematics |pages=275–320 |chapter=Exterior algebra and differential calculus |doi=10.1007/978-1-4684-9461-7_7 |oclc=2401829 |chapter-url=https://link.springer.com/chapter/10.1007/978-1-4684-9461-7_7}}</ref>

Following Grassmann, developments in multilinear algebra were made by [[Victor Schlegel]] in 1872 with the publication of the first part of his ''System der Raumlehre''<ref>{{Cite book |last=Schlegel |first=Victor |title=System der Raumlehre: Nach den Prinzipien der Grassmann'schen Ausdehnungslehre und als Einleitung in Dieselbe; Geometrie; Die Gebiete des Punktes, der Geraden, der Ebene |date=2018 |publisher=Forgotten Books |isbn=978-0-364-22177-8}}</ref> and by [[Elwin Bruno Christoffel]]. Notably, significant advancements came through the work of [[Gregorio Ricci-Curbastro]] and [[Tullio Levi-Civita]],<ref>{{cite journal |last1=Ricci-Curbastro |first1=Gregorio |last2=Levi-Civita |first2=Tullio |year=1900 |title=Méthodes de calcul différentiel absolu et leurs applications |url=https://zenodo.org/record/1428270 |journal=[[Mathematische Annalen]] |volume=54 |issue=1 |pages=125–201 |doi=10.1007/BF01454201 |issn=1432-1807 |s2cid=120009332 |authorlink1=Gregorio Ricci-Curbastro |authorlink2=Tullio Levi-Civita}}</ref> particularly in the form of ''absolute differential calculus'' within multilinear algebra. [[Marcel Grossmann]] and [[Michele Besso]] introduced this form to [[Albert Einstein]], and in 1915, Einstein's publication on [[general relativity]], explaining the [[Precession of the perihelion of Mercury|precession of Mercury's perihelion]], established multilinear algebra and tensors as important mathematical tools in physics.

In 1958, [[Nicolas Bourbaki]] included a chapter on multilinear algebra titled "''Algèbre Multilinéaire''" in his series [[Éléments de mathématique]], specifically within the algebra book. The chapter covers topics such as bilinear functions, the [[tensor product]] of two [[Module (ring theory)|modules]], and the properties of tensor products.<ref>[[Nicolas Bourbaki]] (1958) ''Algèbra Multilinéair'', chapter 3 of book 2 ''Algebra'', in [[Éléments de mathématique]], Paris: Hermann</ref>