This is a 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.
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

1. ^ James & James 1992, p. 7.
2. ^ a b c James & James 1992, p. 27.
3. ^ James & James 1992, p. 66.
4. 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.