Glossary of linear algebra

This is a glossary of linear algebra.

See also: glossary of module theory.

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 ${\displaystyle 1}$ 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 ${\displaystyle 1}$.^[4]

Inverse matrix

Of a matrix ${\displaystyle A}$, another matrix ${\displaystyle B}$ such that ${\displaystyle A}$ multiplied by ${\displaystyle B}$ and ${\displaystyle B}$ multiplied by ${\displaystyle A}$ 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 ${\overrightarrow{v}}_1,\dots ,{\overrightarrow{v}}_n$ is a nonzero tuple of scalar coefficients $c_1,\dots ,c_n$ for which the linear combination $c_1{\overrightarrow{v}}_1+\cdots +c_n{\overrightarrow{v}}_n$ equals $\overrightarrow{0}$.

Linear equation

A polynomial equation of degree one (such as ${\displaystyle x=2y-7}$).^[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.

Row vector

A matrix with only one row.^[4]

Singular-value decomposition

a factorization of an ${\displaystyle m\times n}$ complex matrix M as ${\displaystyle \mathbf{𝐔}\mathrm{\Sigma }{\mathbf{𝐕}}^{\mathbf{*}}}$, where U is an ${\displaystyle m\times m}$ complex unitary matrix, ${\displaystyle \mathrm{\Sigma }}$ is an ${\displaystyle m\times n}$ rectangular diagonal matrix with non-negative real numbers on the diagonal, and V is an ${\displaystyle n\times n}$ 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. 

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