Glossary of linear algebra

This is a glossary of linear algebra.

A edit

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.