# Glossary of linear algebra

This 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 and their representations as vector spaces.
For a glossary related to the generalization of vector spaces through modules, see glossary of module theory.
* A
* B
* C
* D
* E
* F
* G
* H
* I
* J
* K
* L
* M
* N
* O
* P
* Q
* R
* S
* T
* U
* V
* W
* X
* Y
* Z

## 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 vector space, a linearly independent set of vectors spanning the whole vector space.

### basis vector

    An element of a given basis of a vector space.

## C
### column vector

    A matrix with only one column.

### 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.

## 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 basis of a vector space.

### dual space

    The vector space of all linear forms on a given vector space.

## E
### elementary matrix

    Square matrix that differs from the identity matrix 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 v$^→$$_1$, …, v$^→$$_n${\textstyle {\vec {v}}_{1}, \ldots , {\vec {v}}_{n}} is a nonzero tuple of scalar coefficients c$_1$, …, c$_n${\textstyle c_{1}, \ldots , c_{n}} for which the linear combination c$_1$v$^→$$_1$+⋯+c$_n$v$^→$$_n${\textstyle c_{1}{\vec {v}}_{1}+\cdots +c_{n}{\vec {v}}_{n}} equals 0$^→${\textstyle {\vec {0}}}.

### linear equation

    A polynomial equation of degree one (such as $x=2y-7$).

### linear form

    A linear map from a vector space to its field of scalars

### linear independence

    Property of being not linearly dependent.

### 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.

## M
### matrix

    Rectangular arrangement of numbers or other mathematical objects.

## N
### null vector

    1. Another term for an isotropic vector.
    2. Another term for a zero vector.

## 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

    1. A directed quantity, one with both magnitude and direction.
    2. 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.