From Wikipedia, the free encyclopedia
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
vector s 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
{\displaystyle 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 form s 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
{\displaystyle 1}
.
Inverse matrix
Of a matrix
A
{\displaystyle A}
, another matrix
B
{\displaystyle B}
such that
A
{\displaystyle A}
multiplied by
B
{\displaystyle B}
and
B
{\displaystyle B}
multiplied by
A
{\displaystyle 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
v
→
1
,
…
,
v
→
n
{\textstyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n}}
is a nonzero tuple of scalar coefficients
c
1
,
…
,
c
n
{\textstyle c_{1},\ldots ,c_{n}}
for which the linear combination
c
1
v
→
1
+
⋯
+
c
n
v
→
n
{\textstyle c_{1}{\vec {v}}_{1}+\cdots +c_{n}{\vec {v}}_{n}}
equals
0
→
{\textstyle {\vec {0}}}
.
Linear equation
A
polynomial equation of degree one (such as
x
=
2
y
−
7
{\displaystyle 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 space s 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
×
n
{\displaystyle m\times n}
complex matrix M as
U
Σ
V
∗
{\displaystyle \mathbf {U\Sigma V^{*}} }
, where U is an
m
×
m
{\displaystyle m\times m}
complex
unitary matrix ,
Σ
{\displaystyle \mathbf {\Sigma } }
is an
m
×
n
{\displaystyle m\times n}
rectangular diagonal matrix with non-negative real numbers on the diagonal, and V is an
n
×
n
{\displaystyle 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.
Notes
References