Glossary of linear algebra

This is a glossary of linear algebra.

See also: glossary of module theory.

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

B
basis: In a, a set of s spanning the whole vector space. basis vector: An element of a given basis of a vector space.

C
column vector: A with only one column. Coordinate vector: The tuple of the coordinates of a on a. Covector: An element of the of a, (that is a ), identified to an element of the vector space through an inner product.

D
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. dimension: The number of elements of any of a. Dual space: The of all s on a given vector space.

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

I
Identity matrix: A diagonal matrix all of the diagonal elements of which are equal to $1$. 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. 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.

L
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). Linear dependence: A linear dependence of a tuple of vectors $\vec v_1,\ldots,\vec v_n$ is a nonzero tuple of scalar coefficients $c_1,\ldots,c_n$ for which the linear combination $c_1\vec v_1+\cdots+c_n\vec v_n$  equals $\vec0$. Linear equation: A polynomial equation of degree one (such as $x = 2y - 7$). Linear form: A from a to its field of scalars Linear independence: Property of being not. Linear map: A function between s which respects addition and scalar multiplication. Linear transformation: A whose domain and codomain are equal; it is generally supposed to be invertible.

M
matrix: Rectangular arrangement of numbers or other mathematical objects.

N
Null vector: Another term for an. Another term for a.

R
Row vector: A matrix with only one row.

S
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. Spectrum: Set of the eigenvalues of a matrix. Square matrix: A matrix having the same number of rows as columns.

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

V
Vector: A directed quantity, one with both magnitude and direction. An element of a vector space. 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.

Z
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.