Thisglossary of linear algebra is a list of definitions and terms relevant to the field oflinear algebra, the branch of mathematics concerned with linear equations and their representations asvector spaces.

For a glossary related to the generalization of vector spaces throughmodules, seeglossary 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 avector space, alinearly independent set ofvectors spanning the whole vector space.^[2]

basis vector

An element of a givenbasis of a vector space.^[2]

bilinear form

On vector spaceV over fieldK, a bilinear form is a function${\displaystyle B:V\times V\to K}$ that is linear in each variable.

column vector

Amatrix with only one column.^[3]

complex number

An element of a complex plane

complex plane

Alinear algebra over the real numbers withbasis {1, i }, where i is animaginary unit^[4]

coordinate vector

Thetuple of thecoordinates of avector on abasis.

covector

An element of thedual space of avector space, (that is alinear form), identified to an element of the vector space through aninner product.

determinant

The unique scalar function oversquare matrices which is distributive overmatrix multiplication, multilinear in the rows and columns, and takes the value of${\displaystyle 1}$ for theidentity matrix.^[5]

diagonal matrix

A matrix in which only the entries on the main diagonal are non-zero.^[6]

dimension

The number of elements of anybasis of avector space.^[2]

dot product

Given two vectors of the same length, the dot product is the sum of the products of their corresponding indices.

dual space

Thevector space of alllinear forms on a given vector space.^[7]

elementary matrix

Square matrix that differs from theidentity matrix by at most one entry

identity matrix

A diagonal matrix all of the diagonal elements of which are equal to${\displaystyle 1}$.^[6]

imaginary unit

1.  An operator (x, y) → (y, –x), rotating the plane 90° counterclockwise

2.  In alinear algebra, alinear map which when composed with itself produces the negative of the identity

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.^[6]

isotropic vector

In a vector space with aquadratic form, a non-zero vector for which the form is zero.

isotropic quadratic form

A vector space with a quadratic form which has anull vector.

linear algebra

1.  The branch of mathematics that deals with vectors, vector spaces, linear transformations and systems of linear equations.

2.  Avector space that has abinary operation making it aring. This linear algebra is also known as analgebra over a field.^[8]

linear combination

A sum, each of whose summands is an appropriate vector times an appropriate scalar (or ring element).^[9]

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

Apolynomial equation of degree one (such as${\displaystyle x=2y-7}$).^[10]

linear form

Alinear map from avector space to its field of scalars^[11]

linear independence

Property of being notlinearly dependent.^[12]

linear map

Afunction betweenvector spaces which respects addition and scalar multiplication.

linear transformation

Alinear map whosedomain andcodomain are equal; it is generally supposed to beinvertible.

matrix

Rectangular arrangement of numbers or othermathematical objects.^[6] A matrix is writtenA = (a_{i, j}), where a_{i, j} is the entry at row i and column j.

matrix multiplication

If a matrixA has the same number of columns as does matrixB of rows, then a productC =AB may be formed withc_{i, j} equal to thedot product of row i ofA with column j ofB.

null vector

1.  Another term for anisotropic vector.

2.  Another term for azero vector.

orthogonality

Two vectorsu andv are orthogonal with respect to abilinear formB whenB(u,v) = 0.

orthonormality

A set of vectors is orthonormal when they are allunit vectors and are pairwise orthogonal.

orthogonal matrix

A real square matrix with rows (or columns) that form an orthonormal set.

row vector

A matrix with only one row.^[6]

scalar

A scalar is an element of afield used in the definition of avector space.

singular-value decomposition

a factorization of an${\displaystyle m\times n}$ complex matrixM as${\displaystyle \mathbf{U}\mathbf{\Sigma }{\mathbf{V}}^{\mathbf{\ast }}}$, whereU is an${\displaystyle m\times m}$ complexunitary matrix,${\displaystyle \mathbf{\Sigma }}$ is an${\displaystyle m\times n}$rectangular diagonal matrix with non-negative real numbers on the diagonal, andV is an${\displaystyle n\times n}$ complex unitary matrix.^[13]

spectrum

Set of theeigenvalues of a matrix.^[14]

split-complex number

An element of a split-complex plane

split-complex plane

Alinear algebra over the real numbers withbasis {1, j }, where j is ahyperbolic unit

square matrix

A matrix having the same number of rows as columns.^[6]

transpose

The transpose of an n × m matrixM is an m × n matrixM^T obtained by using the rows ofM for the columns ofM^T.

unit vector

a vector in a normed vector space whosenorm is 1, or aEuclidean vector of length one.^[15]

vector

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

2.  An element of a vector space.^[16]

vector space

Aset, whose elements can be added together, and multiplied by elements of afield (this isscalar multiplication); the set must be anabelian group under addition, and the scalar multiplication must be alinear map.^[17]

zero vector

Theadditive identity in a vector space. In anormed vector space, it is the unique vector of norm zero. In aEuclidean vector space, it is the unique vector of length zero.^[18]

• Curtis, Charles W. (1968)Linear Algebra: an introductory approach, second edition,Allyn & Bacon
• Dickson, L. E (1914)Linear Algebras viaInternet Archive
• 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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