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Branch of mathematics
Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b
Linear_algebra
Vectors mapped to 0 by a linear map
Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0. Lay, David C. (2005), Linear Algebra and Its Applications
Kernel_(linear_algebra)
Dimension of the column space of a matrix
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal number
Rank_(linear_algebra)
Sum of elements on the main diagonal
In linear algebra, the trace of a square matrix A, denoted tr(A), is defined as a sum of the elements on its main diagonal, a 11 + a 22 + ⋯ + a n n {\displaystyle
Trace_(linear_algebra)
Algebraic structure in linear algebra
but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrices, which allows computing in vector
Vector_space
Field of mathematics
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which
Numerical_linear_algebra
Idempotent linear transformation from a vector space to itself
In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism)
Projection_(linear_algebra)
In linear algebra, generated subspace
Sheldon Jay (2015). Linear Algebra Done Right (PDF) (3rd ed.). Springer. ISBN 978-3-319-11079-0. Hefferon, Jim (2020). Linear Algebra (PDF) (4th ed.). Orthogonal
Linear_span
Routines for performing common linear algebra operations
Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations
Basic Linear Algebra Subprograms
Basic_Linear_Algebra_Subprograms
Sequence of spaces in linear algebra
In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space V. Here "increasing"
Flag_(linear_algebra)
Set of vectors used to define coordinates
Basis of a matroid Basis of a linear program Coordinate system Change of basis – Coordinate change in linear algebra Frame of a vector space – Similar
Basis_(linear_algebra)
is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector
Outline_of_linear_algebra
Branch of mathematics
variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations. It
Algebra
Sum of terms, each multiplied with a scalar
linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of
Linear_combination
Several equations of degree 1 to be solved simultaneously
-2),} since it makes all three equations valid. Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational
System_of_linear_equations
Undergraduate math course at Harvard University
55b). Previously, the official title was Honors Advanced Calculus and Linear Algebra. The course has gained reputation for its difficulty and accelerated
Math_55
Algebraic structure used in analysis
Lie algebra is the space of all linear maps from a vector space to itself, as discussed below. When the vector space has dimension n, this Lie algebra is
Lie_algebra
Determinant of a subsection of a square matrix
In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix generated from A by removing one or more of its rows and columns
Minor_(linear_algebra)
1966 mathematics textbook by Serge Lang
Linear Algebra is a 1966 mathematics textbook by Serge Lang. The third edition of 1987 covers fundamental concepts of vector spaces, matrices, linear
Linear_Algebra_(book)
In mathematics, vector subspace
specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is
Linear_subspace
Mathematical function, in linear algebra
In mathematics, and more specifically in linear algebra, a linear map, linear mapping, or linear operator is a particular kind of function between vector
Linear_map
Ring that is also a vector space or a module
homomorphism between two R-algebras is an R-linear ring homomorphism. Explicitly, φ : A1 → A2 is an associative algebra homomorphism if φ ( r ⋅ x ) = r ⋅ φ (
Associative_algebra
provide a comparison of linear algebra software libraries, either specialized or general purpose libraries with significant linear algebra coverage. Matrix types
Comparison of linear algebra libraries
Comparison_of_linear_algebra_libraries
Vectors whose linear combinations are nonzero
In linear algebra, a set of vectors is said to be linearly independent if there exists no vector in the set that is equal to a linear combination of the
Linear_independence
Group of 𝑛 × 𝑛 invertible matrices
resulting algebraic structure is a monoid, usually called the full linear monoid, but occasionally also full linear semigroup, general linear monoid etc
General_linear_group
Linear map or polynomial function of degree one
(a linear polynomial). For distinguishing such a linear function from the other concept, the term affine function is often used. In linear algebra, mathematical
Linear_function
Subgroup of the group of invertible n×n matrices
In mathematics, a linear algebraic group is a subgroup of the group of invertible n × n {\displaystyle n\times n} matrices (under matrix multiplication)
Linear_algebraic_group
Branch of mathematics
Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument
Multilinear_algebra
In linear algebra, relation between 3 dimensions
The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity
Rank–nullity_theorem
Concept in mathematics
In mathematics, the special linear Lie algebra of order n {\displaystyle n} over a field F {\displaystyle F} , denoted s l n F {\displaystyle {\mathfrak
Special_linear_Lie_algebra
Vector space equipped with a bilinear product
mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure
Algebra_over_a_field
glossary of linear algebra is a list of definitions and terms relevant to the field of linear algebra, the branch of mathematics concerned with linear equations
Glossary_of_linear_algebra
Algebra associated to any vector space
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle
Exterior_algebra
Similar to the basis of a vector space, but not necessarily linearly independent
In linear algebra, a frame of an inner product space is a generalization of a basis of a vector space to sets that may be linearly dependent. In the terminology
Frame_(linear_algebra)
Vector space consisting of affine subsets
In linear algebra, the quotient of a vector space V {\displaystyle V} by a subspace U {\displaystyle U} is a vector space obtained by "collapsing" U {\displaystyle
Quotient space (linear algebra)
Quotient_space_(linear_algebra)
Array of numbers
a 2 × 3 matrix, or a matrix of dimension 2 × 3. In linear algebra, matrices are used as linear maps. In geometry, matrices are used for geometric transformations
Matrix_(mathematics)
Equivalence under a change of basis (linear algebra)
In linear algebra, two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that B = P − 1 A P . {\displaystyle
Matrix_similarity
Mathematical structure in abstract algebra
conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints. However, it may happen that an algebra admits no involution. Look
*-algebra
Advanced optimization framework for TensorFlow
XLA (Accelerated Linear Algebra) is an open-source compiler for machine learning developed by the OpenXLA project. XLA is designed to improve the performance
Accelerated_Linear_Algebra
Properties of mathematical relationships
transformation). Linear algebra is the branch of mathematics concerned with systems of linear equations. In Boolean algebra, a linear function is a function
Linearity
Branch of mathematics that studies abstract algebraic structures
branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence
Representation_theory
Branch of functional analysis
functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication
Operator_algebra
Professional mathematical society
International Linear Algebra Society (ILAS) is a professional mathematical society organized to promote research and education in linear algebra, matrix theory
International Linear Algebra Society
International_Linear_Algebra_Society
Linear map from a vector space to its field of scalars
(Terse) Introduction to Linear Algebra, American Mathematical Society, ISBN 978-0-8218-4419-9 Lax, Peter (1996), Linear algebra, Wiley-Interscience,
Linear_form
Vector space with generalized dot product
authors, especially in physics and matrix algebra, prefer to define inner products and sesquilinear forms with linearity in the second argument rather than the
Inner_product_space
Academic journal
Linear Algebra and Its Applications is a biweekly peer-reviewed mathematics journal published by Elsevier and covering matrix theory and finite-dimensional
Linear Algebra and Its Applications
Linear_Algebra_and_Its_Applications
Mathematical form
_{i+j=k}a_{i}\cdot b_{j}} There are many different kinds of products in linear algebra. Some of these have confusingly similar names (outer product, exterior
Product_(mathematics)
Basic concepts of algebra
{b^{2}-4ac}}}{2a}}}}}} Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted
Elementary_algebra
Algebra based on a vector space with a quadratic form
mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure
Clifford_algebra
Direct sum of simple Lie algebras
semisimple Lie algebra is a linear Lie algebra under the adjoint representation. This may lead to some ambiguity, as every Lie algebra is already linear with respect
Semisimple_Lie_algebra
Algebraic structure with addition and multiplication
the following application to linear algebra. Let V be a finite-dimensional vector space over a field k and f : V → V a linear map with minimal polynomial
Ring_(mathematics)
Integer matrices with +1 or −1 determinant; invertible over the integers. GL_n(Z)
Fujishige, Satoru (1984), "A System of Linear inequalities with a Submodular Function on (0, ±1) Vectors", Linear Algebra and Its Applications, 63: 253–266
Unimodular_matrix
Type of programming language
more accessible, efficient, and versatile. Linear algebra Mathematical optimization Convex optimization Linear programming Quadratic programming Computational
Scientific programming language
Scientific_programming_language
American mathematician (born 1934)
finite element theory, the calculus of variations, wavelet analysis and linear algebra. He has made many contributions to mathematics education, including
Gilbert_Strang
Open-source linear algebra library
Eigen is a high-level C++ library of template headers for linear algebra, matrix and vector operations, geometrical transformations, numerical solvers
Eigen_(C++_library)
theorem (linear algebra) Bregman–Minc inequality (discrete mathematics) Cauchy-Binet formula (linear algebra) Cayley–Hamilton theorem (Linear algebra) Dimension
List_of_theorems
Software library for numerical linear algebra
LAPACK ("Linear Algebra Package") is a standard software library for numerical linear algebra. It provides routines for solving systems of linear equations
LAPACK
Mathematical prize
The Hans Schneider Prize in Linear Algebra is awarded every three years by the International Linear Algebra Society. It recognizes research, contributions
Hans Schneider Prize in Linear Algebra
Hans_Schneider_Prize_in_Linear_Algebra
Algebraic structure in mathematics
In mathematics, a quadratic algebra is an algebra over a ring for which the algebra extends the ring by a new element that satisfies a monic, quadratic
Quadratic_algebra
Concepts from linear algebra
In linear algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by
Eigenvalues_and_eigenvectors
Real square matrix whose columns and rows are orthogonal unit vectors
In linear algebra, an orthogonal matrix or orthonormal matrix Q, is a real-valued square matrix whose columns and rows are orthonormal vectors. One way
Orthogonal_matrix
Ability of numerical algorithms to remain accurate under small changes of inputs
numerical linear algebra, and another is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra
Numerical_stability
American mathematician (born 1949)
shows how one can teach or learn linear algebra without the use of determinants. Axler later wrote a textbook, Linear Algebra Done Right (4th ed. 2024), to
Sheldon_Axler
Matrix operation which flips a matrix over its diagonal
In linear algebra, transposition is an operation that flips a matrix over its diagonal; that is, transposition switches the row and column indices of the
Transpose
Elements of a field, e.g. real numbers, in the context of linear algebra
In mathematics, more specifically in linear algebra, a scalar is an element of a field which is used to define a vector space through the operation of
Scalar_(mathematics)
Every polynomial has a real or complex root
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Algebraic variety with a group structure
linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry
Algebraic_group
commonly taken from group theory and linear algebra. Algebraic K-theory an important part of homological algebra concerned with defining and applying
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Java software and development tools
statistics, linear algebra, and optimization. JAMA – numerical linear algebra library Jblas: Linear Algebra for Java (Jblas) – linear algebra library using
List of Java software and tools
List_of_Java_software_and_tools
Length in solid geometry
to the origin may be expressed more succinctly using notation from linear algebra. The expression a x + b y + c z {\displaystyle ax+by+cz} in the definition
Distance from a point to a plane
Distance_from_a_point_to_a_plane
Number of vectors in any basis of the vector space
Bridge Connecting Algebra, Modular Forms and Physics, Cambridge University Press, ISBN 978-0-521-83531-2 Axler, Sheldon (2015). Linear Algebra Done Right. Undergraduate
Dimension_(vector_space)
In mathematics, vector space of linear forms
the algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals
Dual_space
these solutions. Pre-algebra Elementary algebra Boolean algebra Abstract algebra Linear algebra Universal algebra An algebraic equation is an equation
Outline_of_algebra
Matrix with nonzero elements on the main diagonal and the diagonals above and below it
In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first
Tridiagonal_matrix
range of requirements such as: desired features (e.g. large dimensional linear algebra, parallel computation, partial differential equations), licensing, readability
List_of_numerical_libraries
Algebraic structure with addition, multiplication, and division
importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number
Field_(mathematics)
ring). *-algebra Affine Lie algebra Akivis algebra Algebra for a monad Albert algebra Alternative algebra AW*-algebra Azumaya algebra Banach algebra Birman–Wenzl
List_of_algebras
In algebra, linear equations and systems of linear equations over a field are widely studied. "Over a field" means that the coefficients of the equations
Linear_equation_over_a_ring
In mathematics, invariant of square matrices
Campbell, H: "Linear Algebra With Applications", pages 111–112. Appleton Century Crofts, 1971 Eves 1990, p. 405 A Brief History of Linear Algebra and Matrix
Determinant
Matrix equal to its transpose
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, A is symmetric ⟺ A = A T . {\displaystyle A{\text{
Symmetric_matrix
Function that is its own inverse
geometry is a polarity that is a correlation of period 2. In linear algebra, an involution is a linear operator T on a vector space, such that T2 = I. Except
Involution_(mathematics)
Mathematical concept in algebra
In linear algebra, a nilpotent matrix is a square matrix N such that N k = 0 {\displaystyle N^{k}=0\,} for some positive integer k {\displaystyle k} .
Nilpotent_matrix
British numerical analyst (1961–2024)
and Applied Mathematics (SIAM), and the 2022 Hans Schneider Prize in Linear Algebra. Higham held a prestigious Royal Society Wolfson Research Merit Award
Nicholas_Higham
Kind of square matrix in linear algebra
In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has
Hessenberg_matrix
Normed vector space that is complete
bounded linear operators on X , {\displaystyle X,} with the composition of maps as product, is a Banach algebra. A C*-algebra is a complex Banach algebra A
Banach_space
Algebraic structure also called skew field
division ring arises in this fashion from some simple module. Much of linear algebra may be formulated, and remains correct, for modules over a division
Division_ring
C++ software for numerical linear algebra
LAPACK++, the Linear Algebra PACKage in C++, is a computer software library of algorithms for numerical linear algebra that solves systems of linear equations
LAPACK++
Induced map between the dual spaces of the two vector spaces
In linear algebra and functional analysis, the transpose or algebraic adjoint of a linear map between two vector spaces, defined over the same field, is
Transpose_of_a_linear_map
Vector space on which a distance is defined
topological space. A straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising
Normed_vector_space
Fourier series of a periodic odd function includes only sine terms. In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose
Symmetry_in_mathematics
Branch of mathematics
approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph
Algebraic_graph_theory
Representation of a matrix as a product
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices
Matrix_decomposition
Series of two mathematics textbooks
of Volume 1 was published in 1967, incorporating two new chapters on linear algebra. Volume 2 received a second edition in 1969. The first volume covers
Calculus_(Apostol_books)
dimensional algebras over fields. Projective cover Radical of a module Socle (mathematics) David Eisenbud, Commutative algebra with a view toward Algebraic Geometry
Top_(algebra)
Describes the objects of a given type, up to some equivalence
descriptions of redirect targetss (by dimension) Rank–nullity theorem – In linear algebra, relation between 3 dimensions (by rank and nullity) Structure theorem
Classification_theorem
Turkish diplomat and mathematician
1897 – August 1898. He is remembered for his Linear Algebra (1882, 1892) which outlined some vector algebra including a "special perpendicular" (cross product)
Hüseyin_Tevfik_Pasha
English mathematician and computer scientist
having received special recognition for his work in computations in linear algebra and 'backward' error analysis." In the same year, he also gave the Society
James_H._Wilkinson
Indian mathematician
Journal of Linear Algebra, Indian Journal of Pure and Applied Mathematics, Kerala Mathematical Association Bulletin, and Linear and Multilinear Algebra. He is
Ravindra_Bapat
Mathematical term
way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is
Adjoint_representation
Series of books published by Springer-Verlag
Intermediate Calculus. ISBN 978-0-387-96058-6. Curtis, Charles W. (1984). Linear Algebra: An Introductory Approach. ISBN 978-0-387-90992-9. Driver, R.D. (1984)
Undergraduate Texts in Mathematics
Undergraduate_Texts_in_Mathematics
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