Glossary of linear algebra

This is a glossary of linear algebra.

A

Affine transformation: A composition of functions consisting of a linear transformation between vector spaces followed by a translation.[1] Equivalently, a function between vector spaces that preserves affine combinations.
Affine combination: A linear combination in which the sum of the coefficients is 1.

Basis: In a vector space, a linearly independent set of vectors spanning the whole vector space.[2]
Basis vector: An element of a given basis of a vector space.[2]

Column vector: A matrix with only one column.[3]
Coordinate vector: The tuple of the coordinates of a vector on a basis.
Covector: An element of the dual space of a vector space, (that is a linear form), identified to an element of the vector space through an inner product.

Determinant: The unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value of $1$ for the unit matrix.
Diagonal matrix: A matrix in which only the entries on the main diagonal are non-zero.[4]
Dimension: The number of elements of any basis of a vector space.[2]
Dual space: The vector space of all linear forms on a given vector space.[5]

Elementary matrix: Square matrix that differs from the identity matrix by at most one entry

Identity matrix: A diagonal matrix all of the diagonal elements of which are equal to $1$ .[4]
Inverse matrix: Of a matrix $A$ , another matrix $B$ such that $A$ multiplied by $B$ and $B$ multiplied by $A$ both equal the identity matrix.[4]
Isotropic vector: In a vector space with a quadratic form, a non-zero vector for which the form is zero.
Isotropic quadratic form: A vector space with a quadratic form which has a null vector.

Linear algebra: The branch of mathematics that deals with vectors, vector spaces, linear transformations and systems of linear equations.
Linear combination: A sum, each of whose summands is an appropriate vector times an appropriate scalar (or ring element).[6]
Linear dependence: A linear dependence of a tuple of vectors ${\textstyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n}}$ is a nonzero tuple of scalar coefficients ${\textstyle c_{1},\ldots ,c_{n}}$ for which the linear combination ${\textstyle c_{1}{\vec {v}}_{1}+\cdots +c_{n}{\vec {v}}_{n}}$ equals ${\textstyle {\vec {0}}}$ .
Linear equation: A polynomial equation of degree one (such as $x=2y-7$ ).[7]
Linear form: A linear map from a vector space to its field of scalars[8]
Linear independence: Property of being not linearly dependent.[9]
Linear map: A function between vector spaces which respects addition and scalar multiplication.
Linear transformation: A linear map whose domain and codomain are equal; it is generally supposed to be invertible.

Matrix: Rectangular arrangement of numbers or other mathematical objects.[4]

Null vector: 1. Another term for an isotropic vector.; 2. Another term for a zero vector.

Singular-value decomposition: a factorization of an $m\times n$ complex matrix $M$ as $\mathbf {U\Sigma V^{*}}$ , where $U$ is an $m\times m$ complex unitary matrix, $\mathbf {\Sigma }$ is an $m\times n$ rectangular diagonal matrix with non-negative real numbers on the diagonal, and $V$ is an $n\times n$ complex unitary matrix.[10]
Spectrum: Set of the eigenvalues of a matrix.[11]
Square matrix: A matrix having the same number of rows as columns.[4]

Unit vector: a vector in a normed vector space whose norm is 1, or a Euclidean vector of length one.[12]

Vector: 1. A directed quantity, one with both magnitude and direction.; 2. An element of a vector space.[13]
Vector space: A set, whose elements can be added together, and multiplied by elements of a field (this is scalar multiplication); the set must be an abelian group under addition, and the scalar multiplication must be a linear map.[14]

Zero vector: The additive identity in a vector space. In a normed vector space, it is the unique vector of norm zero. In a Euclidean vector space, it is the unique vector of length zero.[15]

James, Robert C.; James, Glenn (1992). Mathematics Dictionary (5th ed.). Chapman and Hall. ISBN 978-0442007416.
Bourbaki, Nicolas (1989). Algebra I. Springer. ISBN 978-3540193739.
Williams, Gareth (2014). Linear algebra with applications (8th ed.). Jones & Bartlett Learning.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Linear algebra
Outline Glossary
Basic concepts	Scalar Vector Vector space Scalar multiplication Vector projection Linear span Linear map Linear projection Linear independence Linear combination Basis Change of basis Row and column vectors Row and column spaces Kernel Eigenvalues and eigenvectors Transpose Linear equations
Matrices	Block Decomposition Invertible Minor Multiplication Rank Transformation Cramer's rule Gaussian elimination
Bilinear	Orthogonality Dot product Hadamard product Inner product space Outer product Kronecker product Gram–Schmidt process
Multilinear algebra	Determinant Cross product Triple product Seven-dimensional cross product Geometric algebra Exterior algebra Bivector Multivector Tensor Outermorphism
Vector space constructions	Dual Direct sum Function space Quotient Subspace Tensor product
Numerical	Floating-point Numerical stability Basic Linear Algebra Subprograms Sparse matrix Comparison of linear algebra libraries
Category Mathematics portal Commons Wikibooks Wikiversity