Glossary of linear algebra

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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 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 ${\overrightarrow{v}}_1,\dots ,{\overrightarrow{v}}_n$ 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 $x=2y-7$).

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 $\mathbf{𝐔}\mathrm{\Sigma }{\mathbf{𝐕}}^{\mathbf{*}}$, where U is an $m\times m$ complex unitary matrix, $\mathrm{\Sigma }$ 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.

References

• 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.