Glossary of linear algebra

A

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.

B

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]

C

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.

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 ${\displaystyle 1}$ 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]

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 ${\displaystyle 1}$.^[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.

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).^[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 ${\displaystyle x=2y-7}$).^[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.

M

Matrix

Rectangular arrangement of numbers or other mathematical objects.^[4]

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.^[4]

S

Singular-value decomposition

a factorization of an ${\displaystyle m\times n}$ complex matrix M as ${\displaystyle \mathbf {U\Sigma V^{*}} }$, where U is an ${\displaystyle m\times m}$ complex unitary matrix, ${\displaystyle \mathbf {\Sigma } }$ 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]

U

Unit vector

a vector in a normed vector space whose norm is 1, or a Euclidean vector of length one.^[12]

V

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]

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.^[15]

Notes

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.