Linear algebra is the branch of mathematics concerned with linear equations, linear maps, and their representations in vector spaces and through matrices. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionGiven a linear map between vector spaces, is it an isomorphism, and if not, what are its range and its kernel, the elements it sends to zero. 1 Key DebateWhether linear algebra is best understood through its original nineteenth century form, concrete computation with systems of equations, determinants and matrices, or through the abstract axiomatic vector space that only unified those techniques decades later. The field's basic results and techniques were developed and used for perhaps a hundred years before mathematicians even defined an abstract vector space. 1 Classification
Pure or Applied Linear Algebra
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Source Cauchy-Binet formula (Wikipedia)
Source Cholesky Decomposition (Wikipedia)
Source Cramer's Rule (Wikipedia)
Source Eigenvalues and Eigenvectors (Wikipedia)
Source Levi-Civita Symbol (Wikipedia)
Source Linear Map (Wikipedia)
Source MacMahon's master theorem (Wikipedia)
Source Linear Algebra (Wikipedia)
Source Outer Product (Wikipedia)
Source Principal axis theorem (Wikipedia)
Source Rota's Basis Conjecture (Wikipedia)
Source Schur Complement (Wikipedia)
Source Schur Decomposition (Wikipedia)
Source Semilinear Map (Wikipedia)
Source Sherman-Morrison Formula (Wikipedia)
Source Steinitz exchange lemma, Wikipedia
Source Vector Space (Wikipedia)
Source Woodbury matrix identity (Wikipedia)
Sources
1. Linear Algebra (Wikipedia)
Wikipedialead paragraph
Linear algebra is the branch of mathematics concerning linear equations such as a1x1+...+anxn=b, linear maps ..., and their representations in vector spaces and through matrices.
Linear maps section
An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector.
History section
The leading mathematicians of the nineteenth and early twentieth centuries developed and used most of the basic results and techniques of linear algebra for perhaps a hundred years, without even defining an abstract vector space.
View the Source Vector Space (Wikipedia)
Wikimedia FoundationIncludes: Vector Space, lead paragraphQuote, Includes: Vector Space, lead paragraph
The concept of vector spaces is fundamental for linear algebra, together with the concept of matrices, which allows computing in vector spaces.
Steinitz exchange lemma, Wikipedia
Includes: Steinitz Exchange Lemma, Lead sentenceQuote, Includes: Steinitz Exchange Lemma, Lead sentence
The Steinitz exchange lemma is a theorem in linear algebra concerning bases, dimensionality of a vector space, stating that for an
Woodbury matrix identity (Wikipedia)
Includes: Woodbury Matrix Identity, Lead sentenceQuote, Includes: Woodbury Matrix Identity, Lead sentence
In mathematics, specifically linear algebra, the Woodbury matrix identity, named after Max A.
Principal axis theorem (Wikipedia)
Includes: Principal Axis Theorem, Lead sentenceQuote, Includes: Principal Axis Theorem, Lead sentence
In geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated with a ellipsoid or hyperboloid
Linear Map (Wikipedia)
Includes: Linear Map, Lead sentenceQuote, Includes: Linear Map, Lead sentence
In mathematics, and more specifically in linear algebra, a linear map, linear mapping, or linear operator is a particular kind of
Semilinear Map (Wikipedia)
Includes: Semilinear Map, Lead sentenceQuote, Includes: Semilinear Map, Lead sentence
In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function t
Schur Complement (Wikipedia)
Includes: Schur Complement, Lead sentenceQuote, Includes: Schur Complement, Lead sentence
e Schur complement is a key tool in the fields of linear algebra, the theory of matrices, numerical analysis, and statistics.
Outer Product (Wikipedia)
Includes: Outer Product, Lead sentenceQuote, Includes: Outer Product, Lead sentence
In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the f
Levi-Civita Symbol (Wikipedia)
Includes: Levi-Civita Symbol, Lead sentenceQuote, Includes: Levi-Civita Symbol, Lead sentence
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita
Schur Decomposition (Wikipedia)
Includes: Schur Decomposition, Lead sentenceQuote, Includes: Schur Decomposition, Lead sentence
In linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition.
Cauchy-Binet formula (Wikipedia)
Sherman-Morrison Formula (Wikipedia)
Eigenvalues and Eigenvectors (Wikipedia)
Wikimedia FoundationIncludes: Eigenvalues and Eigenvectors, Lead sentenceView the Source Rota's Basis Conjecture (Wikipedia)
Includes: Rota's Basis Conjecture, Lead sentenceQuote, Includes: Rota's Basis Conjecture, Lead sentence
In linear algebra and matroid theory, Rota's basis conjecture is an unproven conjecture concerning rearrangements of bases, named
Cramer's Rule (Wikipedia)
Wikimedia FoundationIncludes: Cramer's Rule, Lead sentenceQuote, Includes: Cramer's Rule, Lead sentence
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as
MacMahon's master theorem (Wikipedia)
Includes: MacMahon's Master Theorem, Lead sentenceQuote, Includes: MacMahon's Master Theorem, Lead sentence
MMT) is a result in enumerative combinatorics and linear algebra.
Cholesky Decomposition (Wikipedia)
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