In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and more specifically in
linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a
mapping between two
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s that preserves the operations of
vector addition and
scalar multiplication. The same names and the same definition are also used for the more general case of
modules over a
ring; see
Module homomorphism.
If a linear map is a
bijection then it is called a . In the case where
, a linear map is called a linear endomorphism. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that
and
are
real vector spaces (not necessarily with
), or it can be used to emphasize that
is a
function space, which is a common convention in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
. Sometimes the term ''
linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
'' has the same meaning as ''linear map'', while in
analysis it does not.
A linear map from
to
always maps the origin of
to the origin of
. Moreover, it maps
linear subspace
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping'');
* linearity of a ''polynomial''.
An example of a li ...
s in
onto linear subspaces in
(possibly of a lower
dimension); for example, it maps a
plane through the
origin in
to either a plane through the origin in
, a
line through the origin in
, or just the origin in
. Linear maps can often be represented as
matrices, and simple examples include
rotation and reflection linear transformations.
In the language of
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, linear maps are the
morphisms of vector spaces, and they form a category
equivalent to
the one of matrices.
Definition and first consequences
Let
and
be vector spaces over the same
field .
A
function is said to be a ''linear map'' if for any two vectors
and any scalar
the following two conditions are satisfied:
*
Additivity / operation of addition
*
Homogeneity of degree 1 / operation of scalar multiplication
Thus, a linear map is said to be ''operation preserving''. In other words, it does not matter whether the linear map is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication.
By
the associativity of the addition operation denoted as +, for any vectors
and scalars
the following equality holds:
Thus a linear map is one which preserves
linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
s.
Denoting the zero elements of the vector spaces
and
by
and
respectively, it follows that
Let
and
in the equation for homogeneity of degree 1:
A linear map
with
viewed as a one-dimensional vector space over itself is called a
linear functional.
These statements generalize to any left-module
over a ring
without modification, and to any right-module upon reversing of the scalar multiplication.
Examples
* A prototypical example that gives linear maps their name is a function
, of which the
graph is a line through the origin.
* More generally, any
homothety centered in the origin of a vector space is a linear map (here is a scalar).
* The zero map
between two vector spaces (over the same
field) is linear.
* The
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
on any module is a linear operator.
* For real numbers, the map
is not linear.
* For real numbers, the map
is not linear (but is an
affine transformation).
* If
is a
real matrix, then
defines a linear map from
to
by sending a
column vector to the column vector
. Conversely, any linear map between
finite-dimensional vector spaces can be represented in this manner; see the , below.
* If
is an
isometry between real
normed spaces such that
then
is a linear map. This result is not necessarily true for complex normed space.
*
Differentiation defines a linear map from the space of all differentiable functions to the space of all functions. It also defines a
linear operator on the space of all
smooth functions (a linear operator is a
linear endomorphism, that is, a linear map with the same
domain and
codomain). Indeed,
* A definite
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
over some
interval is a linear map from the space of all real-valued integrable functions on to
. Indeed,
* An indefinite
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
(or
antiderivative) with a fixed integration starting point defines a linear map from the space of all real-valued integrable functions on
to the space of all real-valued, differentiable functions on
. Without a fixed starting point, the antiderivative maps to the
quotient space of the differentiable functions by the linear space of constant functions.
* If
and
are finite-dimensional vector spaces over a field , of respective dimensions and , then the function that maps linear maps
to matrices in the way described in (below) is a linear map, and even a
linear isomorphism.
* The
expected value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
(which is in fact a function, and as such an element of a vector space) is linear, as for random variables
and
we have
variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
of a random variable is not linear.
File:Streckung eines Vektors.gif, The function f:\R^2 \to \R^2 with f(x, y) = (2x, y) is a linear map. This function scales the x component of a vector by the factor 2.
File:Streckung der Summe zweier Vektoren.gif, The function f(x, y) = (2x, y) is additive: It does not matter whether vectors are first added and then mapped or whether they are mapped and finally added: f(\mathbf a + \mathbf b) = f(\mathbf a) + f(\mathbf b)
File:Streckung homogenitaet Version 3.gif, The function f(x, y) = (2x, y) is homogeneous: It does not matter whether a vector is first scaled and then mapped or first mapped and then scaled: f(\lambda \mathbf a) = \lambda f(\mathbf a)
Linear extensions
Often, a linear map is constructed by defining it on a subset of a vector space and then to the
linear span of the domain.
Suppose
X and
Y are vector spaces and
f : S \to Y is a
function defined on some subset
S \subseteq X.
Then a '' of
f to
X,'' if it exists, is a linear map
F : X \to Y defined on
X that
extends f[One map F is said to another map f if when f is defined at a point s, then so is F and F(s) = f(s).] (meaning that
F(s) = f(s) for all
s \in S) and takes its values from the codomain of
f.
When the subset
S is a vector subspace of
X then a (
Y-valued) linear extension of
f to all of
X is guaranteed to exist if (and only if)
f : S \to Y is a linear map. In particular, if
f has a linear extension to
\operatorname S, then it has a linear extension to all of
X.
The map
f : S \to Y can be extended to a linear map
F : \operatorname S \to Y if and only if whenever
n > 0 is an integer,
c_1, \ldots, c_n are scalars, and
s_1, \ldots, s_n \in S are vectors such that
0 = c_1 s_1 + \cdots + c_n s_n, then necessarily
0 = c_1 f\left(s_1\right) + \cdots + c_n f\left(s_n\right).
If a linear extension of
f : S \to Y exists then the linear extension
F : \operatorname S \to Y is unique and
F\left(c_1 s_1 + \cdots c_n s_n\right) = c_1 f\left(s_1\right) + \cdots + c_n f\left(s_n\right)
holds for all
n, c_1, \ldots, c_n, and
s_1, \ldots, s_n as above.
If
S is linearly independent then every function
f : S \to Y into any vector space has a linear extension to a (linear) map
\;\operatorname S \to Y (the converse is also true).
For example, if
X = \R^2 and
Y = \R then the assignment
(1, 0) \to -1 and
(0, 1) \to 2 can be linearly extended from the linearly independent set of vectors
S := \ to a linear map on
\operatorname\ = \R^2. The unique linear extension
F : \R^2 \to \R is the map that sends
(x, y) = x (1, 0) + y (0, 1) \in \R^2 to
F(x, y) = x (-1) + y (2) = - x + 2 y.
Every (scalar-valued)
linear functional f defined on a
vector subspace of a real or complex vector space
X has a linear extension to all of
X.
Indeed, the
Hahn–Banach dominated extension theorem even guarantees that when this linear functional
f is dominated by some given
seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
p : X \to \R (meaning that
, f(m), \leq p(m) holds for all
m in the domain of
f) then there exists a linear extension to
X that is also dominated by
p.
Matrices
If
V and
W are
finite-dimensional vector spaces and a
basis is defined for each vector space, then every linear map from
V to
W can be represented by a
matrix. This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if
A is a real
m \times n matrix, then
f(\mathbf x) = A \mathbf x describes a linear map
\R^n \to \R^m (see
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
).
Let
\ be a basis for
V. Then every vector
\mathbf \in V is uniquely determined by the coefficients
c_1, \ldots , c_n in the field
\R:
\mathbf = c_1 \mathbf_1 + \cdots + c_n \mathbf _n.
If
f: V \to W is a linear map,
f(\mathbf) = f(c_1 \mathbf_1 + \cdots + c_n \mathbf_n) = c_1 f(\mathbf_1) + \cdots + c_n f\left(\mathbf_n\right),
which implies that the function ''f'' is entirely determined by the vectors
f(\mathbf _1), \ldots , f(\mathbf _n). Now let
\ be a basis for
W. Then we can represent each vector
f(\mathbf _j) as
f\left(\mathbf_j\right) = a_ \mathbf_1 + \cdots + a_ \mathbf_m.
Thus, the function
f is entirely determined by the values of
a_. If we put these values into an
m \times n matrix
M, then we can conveniently use it to compute the vector output of
f for any vector in
V. To get
M, every column
j of
M is a vector
\begin a_ \\ \vdots \\ a_ \end
corresponding to
f(\mathbf _j) as defined above. To define it more clearly, for some column
j that corresponds to the mapping
f(\mathbf _j),
\mathbf = \begin
\ \cdots & a_ & \cdots\ \\
& \vdots & \\
& a_ &
\end
where
M is the matrix of
f. In other words, every column
j = 1, \ldots, n has a corresponding vector
f(\mathbf _j) whose coordinates
a_, \cdots, a_ are the elements of column
j. A single linear map may be represented by many matrices. This is because the values of the elements of a matrix depend on the bases chosen.
The matrices of a linear transformation can be represented visually:
# Matrix for
T relative to
B:
A
# Matrix for
T relative to
B':
A'
# Transition matrix from
B' to
B:
P
# Transition matrix from
B to
B':
P^
Such that starting in the bottom left corner
\left mathbf\right and looking for the bottom right corner
\left \left(\mathbf\right)\right, one would left-multiply—that is,
A'\left mathbf\right = \left \left(\mathbf\right)\right. The equivalent method would be the "longer" method going clockwise from the same point such that
\left mathbf\right is left-multiplied with
P^AP, or
P^AP\left mathbf\right = \left \left(\mathbf\right)\right.
Examples in two dimensions
In two-
dimensional space R
2 linear maps are described by 2 × 2
matrices. These are some examples:
*
rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
** by 90 degrees counterclockwise:
\mathbf = \begin 0 & -1\\ 1 & 0\end
** by an angle ''θ'' counterclockwise:
\mathbf = \begin \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end
*
reflection
** through the ''x'' axis:
\mathbf = \begin 1 & 0\\ 0 & -1\end
** through the ''y'' axis:
\mathbf = \begin-1 & 0\\ 0 & 1\end
** through a line making an angle ''θ'' with the origin:
\mathbf = \begin\cos2\theta & \sin2\theta \\ \sin2\theta & -\cos2\theta \end
*
scaling by 2 in all directions:
\mathbf = \begin 2 & 0\\ 0 & 2\end = 2\mathbf
*
horizontal shear mapping:
\mathbf = \begin 1 & m\\ 0 & 1\end
* skew of the ''y'' axis by an angle ''θ'':
\mathbf = \begin 1 & -\sin\theta\\ 0 & \cos\theta\end
*
squeeze mapping:
\mathbf = \begin k & 0\\ 0 & \frac\end
*
projection onto the ''y'' axis:
\mathbf = \begin 0 & 0\\ 0 & 1\end.
If a linear map is only composed of rotation, reflection, and/or uniform scaling, then the linear map is a
conformal linear transformation.
Vector space of linear maps
The composition of linear maps is linear: if
f: V \to W and
g: W \to Z are linear, then so is their
composition g \circ f: V \to Z. It follows from this that the
class of all vector spaces over a given field ''K'', together with ''K''-linear maps as
morphisms, forms a
category.
The
inverse of a linear map, when defined, is again a linear map.
If
f_1: V \to W and
f_2: V \to W are linear, then so is their
pointwise sum
f_1 + f_2, which is defined by
(f_1 + f_2)(\mathbf x) = f_1(\mathbf x) + f_2(\mathbf x).
If
f: V \to W is linear and
\alpha is an element of the ground field
K, then the map
\alpha f, defined by
(\alpha f)(\mathbf x) = \alpha (f(\mathbf x)), is also linear.
Thus the set
\mathcal(V, W) of linear maps from
V to
W itself forms a vector space over
K, sometimes denoted
\operatorname(V, W). Furthermore, in the case that
V = W, this vector space, denoted
\operatorname(V), is an
associative algebra under
composition of maps, since the composition of two linear maps is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below.
Given again the finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the
matrix multiplication, the addition of linear maps corresponds to the
matrix addition, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.
Endomorphisms and automorphisms
A linear transformation
f : V \to V is an
endomorphism of
V; the set of all such endomorphisms
\operatorname(V) together with addition, composition and scalar multiplication as defined above forms an
associative algebra with identity element over the field
K (and in particular a
ring). The multiplicative identity element of this algebra is the
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
\operatorname: V \to V.
An endomorphism of
V that is also an
isomorphism is called an
automorphism of
V. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of
V forms a
group, the
automorphism group of
V which is denoted by
\operatorname(V) or
\operatorname(V). Since the automorphisms are precisely those
endomorphisms which possess inverses under composition,
\operatorname(V) is the group of
units in the ring
\operatorname(V).
If
V has finite dimension
n, then
\operatorname(V) is
isomorphic to the
associative algebra of all
n \times n matrices with entries in
K. The automorphism group of
V is
isomorphic to the
general linear group
In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
\operatorname(n, K) of all
n \times n invertible matrices with entries in
K.
Kernel, image and the rank–nullity theorem
If
f: V \to W is linear, we define the
kernel and the
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
or
range of
f by
\begin
\ker(f) &= \ \\
\operatorname(f) &= \
\end
\ker(f) is a
subspace of
V and
\operatorname(f) is a subspace of
W. The following
dimension formula is known as the
rank–nullity theorem:
\dim(\ker( f )) + \dim(\operatorname( f )) = \dim( V ).
The number
\dim(\operatorname(f)) is also called the
rank of
f and written as
\operatorname(f), or sometimes,
\rho(f);
[ p. 52, § 2.5.1][ p. 90, § 50] the number
\dim(\ker(f)) is called the
nullity of
f and written as
\operatorname(f) or
\nu(f).
[ If V and W are finite-dimensional, bases have been chosen and f is represented by the matrix A, then the rank and nullity of f are equal to the rank and nullity of the matrix A, respectively.
]
Cokernel
A subtler invariant of a linear transformation f: V \to W is the ''co''kernel, which is defined as
\operatorname(f) := W/f(V) = W/\operatorname(f).
This is the ''dual'' notion to the kernel: just as the kernel is a ''sub''space of the ''domain,'' the co-kernel is a ''quotient'' space of the ''target.'' Formally, one has the exact sequence
0 \to \ker(f) \to V \to W \to \operatorname(f) \to 0.
These can be interpreted thus: given a linear equation ''f''(v) = w to solve,
* the kernel is the space of ''solutions'' to the ''homogeneous'' equation ''f''(v) = 0, and its dimension is the number of degrees of freedom in the space of solutions, if it is not empty;
* the co-kernel is the space of constraints that the solutions must satisfy, and its dimension is the maximal number of independent constraints.
The dimension of the co-kernel and the dimension of the image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space ''W''/''f''(''V'') is the dimension of the target space minus the dimension of the image.
As a simple example, consider the map ''f'': R2 → R2, given by ''f''(''x'', ''y'') = (0, ''y''). Then for an equation ''f''(''x'', ''y'') = (''a'', ''b'') to have a solution, we must have ''a'' = 0 (one constraint), and in that case the solution space is (''x'', ''b'') or equivalently stated, (0, ''b'') + (''x'', 0), (one degree of freedom). The kernel may be expressed as the subspace (''x'', 0) < ''V'': the value of ''x'' is the freedom in a solution – while the cokernel may be expressed via the map ''W'' → R, (a, b) \mapsto (a): given a vector (''a'', ''b''), the value of ''a'' is the ''obstruction'' to there being a solution.
An example illustrating the infinite-dimensional case is afforded by the map ''f'': R∞ → R∞, \left\ \mapsto \left\ with ''b''1 = 0 and ''b''''n'' + 1 = ''an'' for ''n'' > 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same sum as the rank and the dimension of the co-kernel (\aleph_0 + 0 = \aleph_0 + 1), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an endomorphism have the same dimension (0 ≠ 1). The reverse situation obtains for the map ''h'': R∞ → R∞, \left\ \mapsto \left\ with ''cn'' = ''a''''n'' + 1. Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1.
Index
For a linear operator with finite-dimensional kernel and co-kernel, one may define ''index'' as:
\operatorname(f) := \dim(\ker(f)) - \dim(\operatorname(f)),
namely the degrees of freedom minus the number of constraints.
For a transformation between finite-dimensional vector spaces, this is just the difference dim(''V'') − dim(''W''), by rank–nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Conversely, if mapping from a smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom.
The index of an operator is precisely the Euler characteristic of the 2-term complex 0 → ''V'' → ''W'' → 0. In operator theory, the index of Fredholm operators is an object of study, with a major result being the Atiyah–Singer index theorem.
Algebraic classifications of linear transformations
No classification of linear maps could be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.
Let and denote vector spaces over a field and let be a linear map.
Monomorphism
is said to be '' injective'' or a '' monomorphism'' if any of the following equivalent conditions are true:
# is one-to-one as a map of sets.
#
#
# is monic or left-cancellable, which is to say, for any vector space and any pair of linear maps and , the equation implies .
# is left-invertible, which is to say there exists a linear map such that is the identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
on .
Epimorphism
is said to be '' surjective'' or an '' epimorphism'' if any of the following equivalent conditions are true:
# is onto as a map of sets.
#
# is epic or right-cancellable, which is to say, for any vector space and any pair of linear maps and , the equation implies .
# is right-invertible, which is to say there exists a linear map such that is the identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
on .
Isomorphism
is said to be an '' isomorphism'' if it is both left- and right-invertible. This is equivalent to being both one-to-one and onto (a bijection of sets) or also to being both epic and monic, and so being a bimorphism.
If is an endomorphism, then:
* If, for some positive integer , the -th iterate of , , is identically zero, then is said to be nilpotent.
* If , then is said to be idempotent
* If , where is some scalar, then is said to be a scaling transformation or scalar multiplication map; see scalar matrix.
Change of basis
Given a linear map which is an endomorphism whose matrix is ''A'', in the basis ''B'' of the space it transforms vector coordinates as = ''A'' As vectors change with the inverse of ''B'' (vectors coordinates are contravariant) its inverse transformation is = ''B'' '
Substituting this in the first expression
B\left '\right= AB\left '\right/math>
hence
\left '\right= B^AB\left '\right= A'\left '\right
Therefore, the matrix in the new basis is ''A′'' = ''B''−1''AB'', being ''B'' the matrix of the given basis.
Therefore, linear maps are said to be 1-co- 1-contra- variant objects, or type (1, 1) tensors.
Continuity
A ''linear transformation'' between topological vector spaces, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional.[
1.18 Theorem ''Let \Lambda be a linear functional on a topological vector space . Assume \Lambda \mathbf x \neq 0 for some \mathbf x \in X. Then each of the following four properties implies the other three:''
] An infinite-dimensional domain may have discontinuous linear operators.
An example of an unbounded, hence discontinuous, linear transformation is differentiation on the space of smooth functions equipped with the supremum norm (a function with small values can have a derivative with large values, while the derivative of 0 is 0). For a specific example, converges to 0, but its derivative does not, so differentiation is not continuous at 0 (and by a variation of this argument, it is not continuous anywhere).
Applications
A specific application of linear maps is for geometric transformations, such as those performed in computer graphics
Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...
, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to derivatives; or in relativity, used as a device to keep track of the local transformations of reference frames.
Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques.
See also
*
*
*
*
*
*
*
*
* Category of matrices
* Quasilinearization
Notes
Bibliography
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
{{Authority control
Abstract algebra
Functions and mappings
Transformation (function)