Glossary of linear algebra

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This is a glossary of linear algebra.

See also: glossary of module theory.

## Contents

* 1 A
* 2 B
* 3 C
* 4 D
* 5 E
* 6 I
* 7 L
* 8 M
* 9 N
* 10 R
* 11 S
* 12 U
* 13 V
* 14 Z
* 15 Notes
* 16 References

## 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 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 1.[4]
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.[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 v→1,…,v→n is a nonzero tuple of scalar coefficients c1,…,cn for which the linear combination c1v→1+⋯+cnv→n equals 0→.
Linear equation
A polynomial equation of degree one (such as 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 m×n complex matrix M as 𝐔Σ𝐕*, where U is an m×m complex unitary matrix, Σ is an m×n rectangular diagonal matrix with non-negative real numbers on the diagonal, and V is an n×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

1. ↑ James & James 1992, p. 7.
2. ↑ 2.0 2.1 2.2 James & James 1992, p. 27.
3. ↑ James & James 1992, p. 66.
4. ↑ 4.0 4.1 4.2 4.3 4.4 4.5 James & James 1992, p. 263.
5. ↑ James & James 1992, pp. 80, 135.
6. ↑ James & James 1992, p. 251.
7. ↑ James & James 1992, p. 252.
8. ↑ Bourbaki 1989, p. 232.
9. ↑ James & James 1992, p. 111.
10. ↑ Williams 2014, p. 407.
11. ↑ James & James 1992, p. 389.
12. ↑ James & James 1992, p. 463.
13. ↑ James & James 1992, p. 441.
14. ↑ James & James 1992, p. 442.
15. ↑ James & James 1992, p. 452.

## 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.

* v
* t
* e

Linear algebra

Basic concepts|

* Scalar
* Vector
* Vector space
* Scalar multiplication
* Vector projection
* Linear span
* Linear map
* Linear projection
* Linear independence
* Linear combination
* Basis
* Change of basis
* Row and column vectors
* Row and column spaces
* Orthogonality
* Kernel
* Eigenvalues and eigenvectors
* Outer product
* Inner product space
* Dot product
* Transpose
* Gram–Schmidt process
* Linear equations
* Fundamental theorem

Vector algebra|

* Cross product
* Triple product
* Seven-dimensional cross product

Multilinear algebra|

* Geometric algebra
* Exterior algebra
* Bivector
* Multivector
* Tensor
* Outermorphism

Matrices|

* Block
* Decomposition
* Invertible
* Minor
* Multiplication
* Rank
* Transformation
* Cramer's rule
* Gaussian elimination
* Determinant

Algebraic constructions|

* Dual
* Direct sum
* Function space
* Quotient
* Subspace
* Tensor product

Numerical|

* Floating-point
* MATLAB
* Numerical stability
* Basic Linear Algebra Subprograms (BLAS)
* Sparse matrix
* Comparison of linear algebra libraries
* Comparison of numerical analysis software

* Category
* Outline
* Mathematics portal
* Wikibook
* Wikiversity

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