This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces and through matrices.

Linear equations

Linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...

* System of linear equations *

Determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...

** Minor **

Cauchy–Binet formula In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so ...

Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants ...

Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...

* Gauss–Jordan elimination *

Overcompleteness Overcompleteness is a concept from linear algebra that is widely used in mathematics, computer science, engineering, and statistics (usually in the form of overcomplete frames). It was introduced by R. J. Duffin and A. C. Schaeffer in 1952. Fo ...

* Strassen algorithm

Matrices

Matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

* Matrix addition *

Matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...

Basis transformation matrix In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensiona ...

Characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

* Trace *

Eigenvalue, eigenvector and eigenspace In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...

Cayley–Hamilton theorem In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies ...

Spread of a matrix In mathematics, and more specifically matrix theory, the spread of a matrix is the largest distance in the complex plane between any two eigenvalues of the matrix. Definition Let A be a square matrix with eigenvalues \lambda_1, \ldots, \lambda ...

Jordan normal form In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to ...

** Weyr canonical form *

Rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...

Matrix inversion In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplicat ...

invertible matrix In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplicati ...

Pseudoinverse In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized in ...

* Adjugate *

Transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...

Dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...

Symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...

Orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ...

Skew-symmetric matrix In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a ...

Conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...

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Unitary matrix In linear algebra, a Complex number, complex Matrix (mathematics), square matrix is unitary if its conjugate transpose is also its Invertible matrix, inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, esp ...

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Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...

, Antihermitian matrix * Positive-definite,

positive-semidefinite matrix In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a ...

Pfaffian In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depend on the size of the matrix. The value of this polynomial ...

* Projection *

Spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful b ...

Perron–Frobenius theorem In matrix theory, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive component ...

* List of matrices **

Diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ...

main diagonal In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matrix ...

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Diagonalizable matrix In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...

Triangular matrix In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal ar ...

** Tridiagonal matrix **

Block matrix In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original ma ...

Sparse matrix In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse b ...

Hessenberg matrix 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 zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above t ...

Hessian matrix In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...

Vandermonde matrix In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an matrix :V=\begin 1 & x_1 & x_1^2 & \dots & x_1^\\ 1 & x_2 & x_2^2 & \dots & x_2^\\ 1 & x ...

** Stochastic matrix **

Toeplitz matrix In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix: :\qquad\begin a & b ...

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Circulant matrix In linear algebra, a circulant matrix is a square matrix in which all row vectors are composed of the same elements and each row vector is rotated one element to the right relative to the preceding row vector. It is a particular kind of Toeplit ...

** Hankel matrix **

(0,1)-matrix A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix (mathematics), matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. ...

Matrix decompositions

Matrix decomposition In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of ...

Cholesky decomposition In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for effi ...

LU decomposition In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a ...

QR decomposition In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix ''A'' into a product ''A'' = ''QR'' of an orthogonal matrix ''Q'' and an upper triangular matrix ''R''. QR decom ...

Polar decomposition In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive ...

* Reducing subspace *

Singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is r ...

Higher-order singular value decomposition In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one generalization of the matrix singular value decomposition. It has applications in ...

* Schur decomposition **

Schur complement In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows. Suppose ''p'', ''q'' are nonnegative integers, and suppose ''A'', ''B'', ''C'', ''D'' are respectively ''p'' × ''p'', ''p'' × ''q'', ''q'' ...

** Haynsworth inertia additivity formula

Relations

Matrix equivalence In linear algebra, two rectangular ''m''-by-''n'' matrices ''A'' and ''B'' are called equivalent if :B = Q^ A P for some invertible ''n''-by-''n'' matrix ''P'' and some invertible ''m''-by-''m'' matrix ''Q''. Equivalent matrices represent the sa ...

Matrix congruence In mathematics, two square matrices ''A'' and ''B'' over a field are called congruent if there exists an invertible matrix ''P'' over the same field such that :''P''T''AP'' = ''B'' where "T" denotes the matrix transpose. Matrix congruence ...

Matrix similarity In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that B = P^ A P . Similar matrices represent the same linear map under two (possibly) different bases, with bei ...

Matrix consimilarity In linear algebra, two ''n''-by-''n'' matrix (mathematics), matrices ''A'' and ''B'' are called consimilar if : A = S B \bar^ \, for some invertible n \times n matrix S, where \bar denotes the elementwise complex conjugation. So for real matrices ...

Row equivalence In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two ''m'' × ''n'' matrices are row equivalent if and only if they have the same r ...

Computations

Elementary row operations In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multipl ...

* Householder transformation *

Least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the r ...

, linear least squares * Gram–Schmidt process * Woodbury matrix identity

Vector spaces

Vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

* Linear combination *

Linear span In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characteri ...

Linear independence In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts a ...

Scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...

* Basis **

Change of basis In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are consid ...

Hamel basis In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as componen ...

Cyclic decomposition theorem In mathematics, in linear algebra and functional analysis, a cyclic subspace is a certain special Linear subspace, subspace of a vector space associated with a vector in the vector space and a linear map, linear transformation of the vector space. T ...

* Dimension theorem for vector spaces **

Hamel dimension In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...

* Examples of vector spaces *

Linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...

Shear mapping In plane geometry, a shear mapping is a linear map that displaces each point in a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and goes through the origin. This type of map ...

Galilean transformation In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotati ...

** Squeeze mapping or

Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...

* Linear subspace **

Row and column spaces In linear algebra, the column space (also called the range or image) of a matrix ''A'' is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matri ...

Column space In linear algebra, the column space (also called the range or image) of a matrix ''A'' is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding mat ...

** Row space ** Cyclic subspace **

Null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel ...

, nullity **

Rank–nullity theorem The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Th ...

** Nullity theorem *

Dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...

Linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...

Linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , th ...

Category of vector spaces In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ri ...

Structures

Topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...

Normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "leng ...

Inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

Orthogonality In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings i ...

Orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...

Orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if i ...

Orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...

Pseudo-Euclidean space In mathematics and theoretical physics, a pseudo-Euclidean space is a finite- dimensional real -space together with a non-degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving q(x ...

Null vector In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms a ...

Indefinite orthogonal group In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where . It is also called the ps ...

Orientation (geometry) In geometry, the orientation, angular position, attitude, bearing, or direction of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it occupies. More specifically, it refers to the imagi ...

Improper rotation In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendic ...

* Symplectic structure

Multilinear algebra

Multilinear algebra Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of '' ...

Tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...

** Classical treatment of tensors ** Component-free treatment of tensors **

Gamas's Theorem Gamas's theorem is a result in multilinear algebra which states the necessary and sufficient conditions for a tensor symmetrized by an irreducible representation of the symmetric group S_n to be zero. It was proven in 1988 by Carlos Gamas. Addi ...

Outer product In linear algebra, the outer product of two coordinate vectors is a matrix. If the two vectors have dimensions ''n'' and ''m'', then their outer product is an ''n'' × ''m'' matrix. More generally, given two tensors (multidimensional arrays of n ...

Tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of bein ...

Exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...

Symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universa ...

Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...

Geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the g ...

Topics related to affine spaces

Affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...

Affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generall ...

Affine group In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself. It is a Lie group if is the real or complex field or quaternions. ...

Affine geometry In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of '' parallel lines'' is one of the main properties that is i ...

Affine coordinate system In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...

* Flat (geometry) *

Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured ...

Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformation ...

Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...

Galilean group In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotati ...

Projective space

Projective space *

Projective transformation In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In genera ...

Projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...

* Projective linear group *

Quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections ( ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is ...

and

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...