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Linear algebra is the branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
concerning
linear equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

linear equation
s such as: :a_1x_1+\cdots +a_nx_n=b,
linear map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

linear map
s such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s and through
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
, including for defining basic objects such as
lines Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat A long terminal repeat (LTR) is a pair of identical sequences of DNA ...

lines
, planes and
rotations A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...
. Also,
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

engineering
, because it allows
modeling In general, a model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. ...
many natural phenomena, and computing efficiently with such models. For
nonlinear system In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a
multivariate function In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (math ...
at a point is the linear map that best approximates the function near that point.


History

The procedure for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: ''Rectangular Arrays'' of ''
The Nine Chapters on the Mathematical Art ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surv ...
''. Its use is illustrated in eighteen problems, with two to five equations. Systems of linear equations arose in Europe with the introduction in 1637 by
René Descartes René Descartes ( or ; ; Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

René Descartes
of
coordinates In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

coordinates
in
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
. In fact, in this new geometry, now called
Cartesian geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
, lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations. The first systematic methods for solving linear systems used
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

determinant
s, first considered by
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...

Leibniz
in 1693. In 1750,
Gabriel Cramer Gabriel Cramer (; 31 July 1704 – 4 January 1752) was a Genevan mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quan ...

Gabriel Cramer
used them for giving explicit solutions of linear systems, now called
Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical s ...
. Later,
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

Gauss
further described the method of elimination, which was initially listed as an advancement in
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's geometric shape, orientation in space, and gravitational field. The field also incorporates studies of how these properties change over time and equivalent measu ...
. In 1844
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath, known in his day as a linguistics, linguist and now also as a mathematics, mathematician. He was also a physics, physicist, gener ...
published his "Theory of Extension" which included foundational new topics of what is today called linear algebra. In 1848,
James Joseph Sylvester James Joseph Sylvester (3 September 1814 – 15 March 1897) was an United Kingdom, English mathematician. He made fundamental contributions to Matrix (mathematics), matrix theory, invariant theory, number theory, Integer partition, partitio ...

James Joseph Sylvester
introduced the term ''matrix'', which is Latin for ''womb''. Linear algebra grew with ideas noted in the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. For instance, two numbers ''w'' and ''z'' in \mathbb have a difference ''w'' – ''z'', and the line segments \overline and \overline are of the same length and direction. The segments are equipollent. The four-dimensional system \mathbb of
quaternion In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

quaternion
s was started in 1843. The term ''vector'' was introduced as ''v'' = ''x'' i + ''y'' j + ''z'' k representing a point in space. The quaternion difference ''p'' – ''q'' also produces a segment equipollent to \overline . Other
hypercomplex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
systems also used the idea of a linear space with a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
.
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

Arthur Cayley
introduced
matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

matrix multiplication
and the
inverse matrixIn linear algebra, an ''n''-by-''n'' square matrix is called invertible (also nonsingular or nondegenerate), if there exists an ''n''-by-''n'' square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the ''n''-by-''n'' identit ...
in 1856, making possible the
general linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. The mechanism of
group representation In the mathematical field of representation theory Representation theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote a matrix, thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants".
Benjamin Peirce Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

Benjamin Peirce
published his ''Linear Associative Algebra'' (1872), and his son
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, ian, mathematician and scientist who is sometimes known as "the father of ". He was known as a somewhat unusual character. Educated as a chemist an ...

Charles Sanders Peirce
extended the work later. The
telegraph Telegraphy is the long-distance transmission of messages where the sender uses symbolic codes, known to the recipient, rather than a physical exchange of an object bearing the message. Thus flag semaphore Flag semaphore (from the Ancient ...

telegraph
required an explanatory system, and the 1873 publication of
A Treatise on Electricity and Magnetism ''A Treatise on Electricity and Magnetism'' is a two-volume treatise on electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric ...
instituted a field theory of forces and required
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
for expression. Linear algebra is flat differential geometry and serves in tangent spaces to
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
s. Electromagnetic symmetries of spacetime are expressed by the
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the ...

Lorentz transformation
s, and much of the history of linear algebra is the
history of Lorentz transformations The history of Lorentz transformation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the re ...
. The first modern and more precise definition of a vector space was introduced by
Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
in 1888; by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
. The development of computers led to increased research in efficient
algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

algorithm
s for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations.


Vector spaces

Until the 19th century, linear algebra was introduced through systems of linear equations and
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
. In modern mathematics, the presentation through ''vector spaces'' is generally preferred, since it is more
syntheticA synthetic is an artificial material produced by organic chemistry, organic chemical synthesis. Synthetic may also refer to: In the sense of both "combination" and "artificial" * Synthetic chemical or synthetic compress, produced by the process ...
, more general (not limited to the finite-dimensional case), and conceptually simpler, although more abstract. A vector space over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
(often the field of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s) is a set equipped with two
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s satisfying the following
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

axiom
s. Elements of are called ''vectors'', and elements of ''F'' are called ''scalars''. The first operation, ''
vector addition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

vector addition
'', takes any two vectors and and outputs a third vector . The second operation, ''
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module (mathematics), module in abstract algebra). In common geometrical contexts, scalar multiplication of a re ...

scalar multiplication
'', takes any scalar and any vector and outputs a new . The axioms that addition and scalar multiplication must satisfy are the following. (In the list below, and are arbitrary elements of , and and are arbitrary scalars in the field .) The first four axioms mean that is an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
under addition. An element of a specific vector space may have various nature; for example, it could be a
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

sequence
, a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, a
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
or a
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
. Linear algebra is concerned with those properties of such objects that are common to all vector spaces.


Linear maps

Linear maps are mappings between vector spaces that preserve the vector-space structure. Given two vector spaces and over a field , a linear map (also called, in some contexts, linear transformation or linear mapping) is a
map A map is a symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...
: T:V\to W that is compatible with addition and scalar multiplication, that is : T(\mathbf u + \mathbf v)=T(\mathbf u)+T(\mathbf v), \quad T(a \mathbf v)=aT(\mathbf v) for any vectors in and scalar in . This implies that for any vectors in and scalars in , one has : T(a \mathbf u + b \mathbf v)= T(a \mathbf u) + T(b \mathbf v) = aT(\mathbf u) + bT(\mathbf v) When are the same vector space, a linear map T:V\to V is also known as a ''linear operator'' on . A
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
linear map between two vector spaces (that is, every vector from the second space is associated with exactly one in the first) is an
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
. Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view, in the sense that they cannot be distinguished by using vector space properties. An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its
range Range may refer to: Geography * Range (geographic)A range, in geography, is a chain of hill A hill is a landform A landform is a natural or artificial feature of the solid surface of the Earth or other planetary body. Landforms together ...
(or image) and the set of elements that are mapped to the zero vector, called the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of the map. All these questions can be solved by using Gaussian elimination or some variant of this
algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

algorithm
.


Subspaces, span, and basis

The study of those subsets of vector spaces that are in themselves vector spaces under the induced operations is fundamental, similarly as for many mathematical structures. These subsets are called
linear subspace In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s. More precisely, a linear subspace of a vector space over a field is a
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
of such that and are in , for every , in , and every in . (These conditions suffice for implying that is a vector space.) For example, given a linear map T:V\to W , the
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of , and the
inverse image In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

inverse image
of 0 (called
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
or null space), are linear subspaces of and , respectively. Another important way of forming a subspace is to consider
linear combination In mathematics, a linear combination 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'' and ''y'' would be ...
s of a set of vectors: the set of all sums : a_1 \mathbf v_1 + a_2 \mathbf v_2 + \cdots + a_k \mathbf v_k, where are in , and are in form a linear subspace called the span of . The span of is also the intersection of all linear subspaces containing . In other words, it is the smallest (for the inclusion relation) linear subspace containing . A set of vectors is
linearly independent In the theory of vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...
if none is in the span of the others. Equivalently, a set of vectors is linearly independent if the only way to express the zero vector as a linear combination of elements of is to take zero for every coefficient a_i. A set of vectors that spans a vector space is called a
spanning set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
or
generating set In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set (mathematics), set of objects, together with a set of Operation (mathe ...
. If a spanning set is ''linearly dependent'' (that is not linearly independent), then some element of is in the span of the other elements of , and the span would remain the same if one remove from . One may continue to remove elements of until getting a ''linearly independent spanning set''. Such a linearly independent set that spans a vector space is called a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
of . The importance of bases lies in the fact that they are simultaneously minimal generating sets and maximal independent sets. More precisely, if is a linearly independent set, and is a spanning set such that S\subseteq T, then there is a basis such that S\subseteq B\subseteq T. Any two bases of a vector space have the same
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, which is called the
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
of ; this is the
dimension theorem for vector spaces In mathematics, the dimension theorem for vector spaces states that all Basis (linear algebra), bases of a vector space have equally many elements. This number of elements may be finite or infinite (in the latter case, it is a cardinal number), and ...
. Moreover, two vector spaces over the same field are
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

isomorphic
if and only if they have the same dimension. If any basis of (and therefore every basis) has a finite number of elements, is a ''finite-dimensional vector space''. If is a subspace of , then . In the case where is finite-dimensional, the equality of the dimensions implies . If ''U''1 and ''U''2 are subspaces of ''V'', then :\dim(U_1 + U_2) = \dim U_1 + \dim U_2 - \dim(U_1 \cap U_2), where U_1+U_2 denotes the span of U_1\cup U_2.


Matrices

Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps. Their theory is thus an essential part of linear algebra. Let be a finite-dimensional vector space over a field , and be a basis of (thus is the dimension of ). By definition of a basis, the map :\begin (a_1, \ldots, a_m)&\mapsto a_1 \mathbf v_1+\cdots a_m \mathbf v_m\\ F^m &\to V \end is a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijection
from F^m, the set of the
sequences In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of elements of , onto . This is an
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
of vector spaces, if F^m is equipped of its standard structure of vector space, where vector addition and scalar multiplication are done component by component. This isomorphism allows representing a vector by its
inverse image In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

inverse image
under this isomorphism, that is by the coordinates vector (a_1, \ldots, a_m) or by the
column matrixIn linear algebra, a column vector is a column of entries, for example, :\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end \,. Similarly, a row vector is a row of entries, p. 8 :\boldsymbol a = \begin a_1 & a_2 & \dots & a_n \end \,. Throu ...
:\begina_1\\\vdots\\a_m\end. If is another finite dimensional vector space (possibly the same), with a basis (\mathbf w_1, \ldots, \mathbf w_n), a linear map from to is well defined by its values on the basis elements, that is (f(\mathbf w_1), \ldots, f(\mathbf w_n)). Thus, is well represented by the list of the corresponding column matrices. That is, if :f(w_j)=a_v_1 + \cdots+a_v_m, for , then is represented by the matrix :\begin a_&\cdots&a_\\ \vdots&\ddots&\vdots\\ a_&\cdots&a_ \end, with rows and columns.
Matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

Matrix multiplication
is defined in such a way that the product of two matrices is the matrix of the
composition Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing the result of applying the represented linear map to the represented vector. It follows that the theory of finite-dimensional vector spaces and the theory of matrices are two different languages for expressing exactly the same concepts. Two matrices that encode the same linear transformation in different bases are called similar. It can be proved that two matrices are similar if and only if one can transform one into the other by elementary row and column operations. For a matrix representing a linear map from to , the row operations correspond to change of bases in and the column operations correspond to change of bases in . Every matrix is similar to an
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

identity matrix
possibly bordered by zero rows and zero columns. In terms of vector spaces, this means that, for any linear map from to , there are bases such that a part of the basis of is mapped bijectively on a part of the basis of , and that the remaining basis elements of , if any, are mapped to zero. Gaussian elimination is the basic algorithm for finding these elementary operations, and proving these results.


Linear systems

A finite set of linear equations in a finite set of variables, for example, x_1, x_2, \ldots, x_n or x, y, \ldots, z is called a system of linear equations or a linear system. Systems of linear equations form a fundamental part of linear algebra. Historically, linear algebra and matrix theory has been developed for solving such systems. In the modern presentation of linear algebra through vector spaces and matrices, many problems may be interpreted in terms of linear systems. For example, let be a linear system. To such a system, one may associate its matrix :M = \left begin 2 & 1 & -1\\ -3 & -1 & 2 \\ -2 & 1 & 2 \end\right and its right member vector :\mathbf = \begin 8\\-11\\-3 \end. Let be the linear transformation associated to the matrix . A solution of the system () is a vector :\mathbf=\begin x\\y\\z \end such that :T(\mathbf) = \mathbf, that is an element of the
preimage In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of by . Let () be the associated homogeneous system, where the right-hand sides of the equations are put to zero: The solutions of () are exactly the elements of the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of or, equivalently, . The Gaussian-elimination consists of performing
elementary row operationIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s on the
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:\left !\beginM&\mathbf\end\!\right= \left begin 2 & 1 & -1&8\\ -3 & -1 & 2&-11 \\ -2 & 1 & 2&-3 \end\right for putting it in
reduced row echelon form In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form means that Gaussian ...
. These row operations do not change the set of solutions of the system of equations. In the example, the reduced echelon form is :\left !\beginM&\mathbf\end\!\right= \left begin 1 & 0 & 0&2\\ 0 & 1 & 0&3 \\ 0 & 0 & 1&-1 \end\right showing that the system () has the unique solution :\beginx&=2\\y&=3\\z&=-1.\end It follows from this matrix interpretation of linear systems that the same methods can be applied for solving linear systems and for many operations on matrices and linear transformations, which include the computation of the
ranks Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * Hi ...
, kernels,
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s.


Endomorphisms and square matrices

A linear
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group ...
is a linear map that maps a vector space to itself. If has a basis of elements, such an endomorphism is represented by a square matrix of size . With respect to general linear maps, linear endomorphisms and square matrices have some specific properties that make their study an important part of linear algebra, which is used in many parts of mathematics, including
geometric transformation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s,
coordinate change In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s,
quadratic form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, and many other part of mathematics.


Determinant

The ''determinant'' of a square matrix is defined to be :\sum_ (-1)^ a_ \cdots a_, where S_n is the group of all permutations of elements, \sigma is a permutation, and (-1)^\sigma the parity of the permutation. A matrix is invertible if and only if the determinant is invertible (i.e., nonzero if the scalars belong to a field).
Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical s ...
is a
closed-form expression In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, in terms of determinants, of the solution of a system of linear equations in unknowns. Cramer's rule is useful for reasoning about the solution, but, except for or , it is rarely used for computing a solution, since Gaussian elimination is a faster algorithm. The ''determinant of an endomorphism'' is the determinant of the matrix representing the endomorphism in terms of some ordered basis. This definition makes sense, since this determinant is independent of the choice of the basis.


Eigenvalues and eigenvectors

If is a linear endomorphism of a vector space over a field , an ''eigenvector'' of is a nonzero vector of such that for some scalar in . This scalar is an ''eigenvalue'' of . If the dimension of is finite, and a basis has been chosen, and may be represented, respectively, by a square matrix and a column matrix ; the equation defining eigenvectors and eigenvalues becomes :Mz=az. Using the
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

identity matrix
, whose entries are all zero, except those of the main diagonal, which are equal to one, this may be rewritten :(M-aI)z=0. As is supposed to be nonzero, this means that is a
singular matrixIn linear algebra, an ''n''-by-''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
, and thus that its determinant \det(M-aI) equals zero. The eigenvalues are thus the
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of the
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
:\det(xI-M). If is of dimension , this is a
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of degree , called the
characteristic polynomial In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...
of the matrix (or of the endomorphism), and there are, at most, eigenvalues. If a basis exists that consists only of eigenvectors, the matrix of on this basis has a very simple structure: it is a
diagonal matrix In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
such that the entries on the
main diagonal In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
are eigenvalues, and the other entries are zero. In this case, the endomorphism and the matrix are said to be
diagonalizable In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrixIn linear algebra, an ''n''-by-''n'' square matrix is called invertible (also ...
. More generally, an endomorphism and a matrix are also said diagonalizable, if they become diagonalizable after extending the field of scalars. In this extended sense, if the characteristic polynomial is
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, then the matrix is diagonalizable. A
symmetric matrix In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...
is always diagonalizable. There are non-diagonalizable matrices, the simplest being :\begin0&1\\0&0\end (it cannot be diagonalizable since its square is the
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, and the square of a nonzero diagonal matrix is never zero). When an endomorphism is not diagonalizable, there are bases on which it has a simple form, although not as simple as the diagonal form. The Frobenius normal form does not need of extending the field of scalars and makes the characteristic polynomial immediately readable on the matrix. The
Jordan normal form In linear algebra, a Jordan normal form, also known as a Jordan canonical form or JCF, is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to s ...
requires to extend the field of scalar for containing all eigenvalues, and differs from the diagonal form only by some entries that are just above the main diagonal and are equal to 1.


Duality

A
linear form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

linear form
is a linear map from a vector space V over a field F to the field of scalars F, viewed as a vector space over itself. Equipped by
pointwiseIn mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined o ...
addition and multiplication by a scalar, the linear forms form a vector space, called the dual space of V, and usually denoted V^* or V'. If \mathbf v_1, \ldots, \mathbf v_n is a basis of V (this implies that is finite-dimensional), then one can define, for , a linear map v_i^* such that v_i^*(\mathbf e_i)=1 and v_i^*(\mathbf e_j)=0 if . These linear maps form a basis of V^*, called the
dual basis In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
of \mathbf v_1, \ldots, \mathbf v_n. (If is not finite-dimensional, the v^*_i may be defined similarly; they are linearly independent, but do not form a basis.) For \mathbf v in V, the map :f\to f(\mathbf v) is a linear form on V^*. This defines the canonical linear map from V into V^, the dual of V^*, called the bidual of V. This canonical map is an
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
if V is finite-dimensional, and this allows identifying V with its bidual. (In the infinite dimensional case, the canonical map is injective, but not surjective.) There is thus a complete symmetry between a finite-dimensional vector space and its dual. This motivates the frequent use, in this context, of the
bra–ket notation In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum ...
:\langle f, \mathbf x\rangle for denoting f(\mathbf x).


Dual map

Let :f:V\to W be a linear map. For every linear form on , the composite function is a linear form on . This defines a linear map :f^*:W^*\to V^* between the dual spaces, which is called the dual or the transpose of . If and are finite dimensional, and is the matrix of in terms of some ordered bases, then the matrix of f^* over the dual bases is the
transpose In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...

transpose
M^\mathsf T of , obtained by exchanging rows and columns. If elements of vector spaces and their duals are represented by column vectors, this duality may be expressed in
bra–ket notation In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum ...
by :\langle h^\mathsf T , M \mathbf v\rangle = \langle h^\mathsf T M, \mathbf v\rangle. For highlighting this symmetry, the two members of this equality are sometimes written :\langle h^\mathsf T \mid M \mid \mathbf v\rangle.


Inner-product spaces

Besides these basic concepts, linear algebra also studies vector spaces with additional structure, such as an
inner product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. The inner product is an example of a
bilinear form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, and it gives the vector space a geometric structure by allowing for the definition of length and angles. Formally, an ''inner product'' is a map : \langle \cdot, \cdot \rangle : V \times V \to F that satisfies the following three
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

axiom
s for all vectors u, v, w in ''V'' and all scalars ''a'' in ''F'': *
Conjugate Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
symmetry: *::\langle \mathbf u, \mathbf v\rangle =\overline. *:In R, it is symmetric. *
Linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...

Linear
ity in the first argument: *::\langle a \mathbf u, \mathbf v\rangle= a \langle \mathbf u, \mathbf v\rangle. *::\langle \mathbf u + \mathbf v, \mathbf w\rangle= \langle \mathbf u, \mathbf w\rangle+ \langle \mathbf v, \mathbf w\rangle. * Definite bilinear form, Positive-definiteness: *::\langle \mathbf v, \mathbf v\rangle \geq 0 with equality only for v = 0. We can define the length of a vector v in ''V'' by :\, \mathbf v\, ^2=\langle \mathbf v, \mathbf v\rangle, and we can prove the Cauchy–Schwarz inequality: :, \langle \mathbf u, \mathbf v\rangle, \leq \, \mathbf u\, \cdot \, \mathbf v\, . In particular, the quantity :\frac \leq 1, and so we can call this quantity the cosine of the angle between the two vectors. Two vectors are orthogonal if \langle \mathbf u, \mathbf v\rangle =0. An orthonormal basis is a basis where all basis vectors have length 1 and are orthogonal to each other. Given any finite-dimensional vector space, an orthonormal basis could be found by the Gram–Schmidt procedure. Orthonormal bases are particularly easy to deal with, since if , then a_i = \langle \mathbf v, \mathbf v_i \rangle. The inner product facilitates the construction of many useful concepts. For instance, given a transform ''T'', we can define its Hermitian conjugate ''T*'' as the linear transform satisfying : \langle T \mathbf u, \mathbf v \rangle = \langle \mathbf u, T^* \mathbf v\rangle. If ''T'' satisfies ''TT*'' = ''T*T'', we call ''T'' Normal matrix, normal. It turns out that normal matrices are precisely the matrices that have an orthonormal system of eigenvectors that span ''V''.


Relationship with geometry

There is a strong relationship between linear algebra and
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geometry
, which started with the introduction by
René Descartes René Descartes ( or ; ; Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

René Descartes
, in 1637, of Cartesian coordinates. In this new (at that time) geometry, now called
Cartesian geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
, points are represented by Cartesian coordinates, which are sequences of three real numbers (in the case of the usual three-dimensional space). The basic objects of geometry, which are
lines Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat A long terminal repeat (LTR) is a pair of identical sequences of DNA ...

lines
and planes are represented by linear equations. Thus, computing intersections of lines and planes amounts to solving systems of linear equations. This was one of the main motivations for developing linear algebra. Most
geometric transformation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, such as translations, rotations, reflection (mathematics), reflections, rigid motions, isometries, and projection (mathematics), projections transform lines into lines. It follows that they can be defined, specified and studied in terms of linear maps. This is also the case of homography, homographies and Möbius transformations, when considered as transformations of a projective space. Until the end of 19th century, geometric spaces were defined by
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

axiom
s relating points, lines and planes (synthetic geometry). Around this date, it appeared that one may also define geometric spaces by constructions involving vector spaces (see, for example, Projective space and Affine space). It has been shown that the two approaches are essentially equivalent.Emil Artin (1957) ''Geometric Algebra (book), Geometric Algebra'' Interscience Publishers In classical geometry, the involved vector spaces are vector spaces over the reals, but the constructions may be extended to vector spaces over any field, allowing considering geometry over arbitrary fields, including finite fields. Presently, most textbooks, introduce geometric spaces from linear algebra, and geometry is often presented, at elementary level, as a subfield of linear algebra.


Usage and applications

Linear algebra is used in almost all areas of mathematics, thus making it relevant in almost all scientific domains that use mathematics. These applications may be divided into several wide categories.


Geometry of ambient space

The Mathematical model, modeling of ambient space is based on
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
. Sciences concerned with this space use geometry widely. This is the case with mechanics and robotics, for describing rigid body dynamics;
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's geometric shape, orientation in space, and gravitational field. The field also incorporates studies of how these properties change over time and equivalent measu ...
for describing Earth shape; perspectivity, computer vision, and computer graphics, for describing the relationship between a scene and its plane representation; and many other scientific domains. In all these applications, synthetic geometry is often used for general descriptions and a qualitative approach, but for the study of explicit situations, one must compute with
coordinates In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

coordinates
. This requires the heavy use of linear algebra.


Functional analysis

Functional analysis studies function spaces. These are vector spaces with additional structure, such as Hilbert spaces. Linear algebra is thus a fundamental part of functional analysis and its applications, which include, in particular, quantum mechanics (wave functions).


Study of complex systems

Most physical phenomena are modeled by partial differential equations. To solve them, one usually decomposes the space in which the solutions are searched into small, mutually interacting Discretization, cells. For linear systems this interaction involves linear functions. For nonlinear systems, this interaction is often approximated by linear functions. In both cases, very large matrices are generally involved. Weather forecasting is a typical example, where the whole Earth atmosphere is divided in cells of, say, 100 km of width and 100 m of height.


Scientific computation

Nearly all scientific computations involve linear algebra. Consequently, linear algebra algorithms have been highly optimized. Basic Linear Algebra Subprograms, BLAS and LAPACK are the best known implementations. For improving efficiency, some of them configure the algorithms automatically, at run time, for adapting them to the specificities of the computer (cache (computing), cache size, number of available multi-core processor, cores, ...). Some Processor (computing), processors, typically graphics processing units (GPU), are designed with a matrix structure, for optimizing the operations of linear algebra.


Extensions and generalizations

This section presents several related topics that do not appear generally in elementary textbooks on linear algebra, but are commonly considered, in advanced mathematics, as parts of linear algebra.


Module theory

The existence of multiplicative inverses in fields is not involved in the axioms defining a vector space. One may thus replace the field of scalars by a ring (mathematics), ring , and this gives a structure called module over , or -module. The concepts of linear independence, span, basis, and linear maps (also called module homomorphisms) are defined for modules exactly as for vector spaces, with the essential difference that, if is not a field, there are modules that do not have any basis. The modules that have a basis are the free modules, and those that are spanned by a finite set are the finitely generated modules. Module homomorphisms between finitely generated free modules may be represented by matrices. The theory of matrices over a ring is similar to that of matrices over a field, except that
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

determinant
s exist only if the ring is commutative ring, commutative, and that a square matrix over a commutative ring is invertible only if its determinant has a multiplicative inverse in the ring. Vector spaces are completely characterized by their dimension (up to an isomorphism). In general, there is not such a complete classification for modules, even if one restricts oneself to finitely generated modules. However, every module is a cokernel of a homomorphism of free modules. Modules over the integers can be identified with
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s, since the multiplication by an integer may identified to a repeated addition. Most of the theory of abelian groups may be extended to modules over a principal ideal domain. In particular, over a principal ideal domain, every submodule of a free module is free, and the fundamental theorem of finitely generated abelian groups may be extended straightforwardly to finitely generated modules over a principal ring. There are many rings for which there are algorithms for solving linear equations and systems of linear equations. However, these algorithms have generally a computational complexity that is much higher than the similar algorithms over a field. For more details, see Linear equation over a ring.


Multilinear algebra and tensors

In multilinear algebra, one considers multivariable linear transformations, that is, mappings that are linear in each of a number of different variables. This line of inquiry naturally leads to the idea of the dual space, the vector space ''V'' consisting of linear maps where ''F'' is the field of scalars. Multilinear maps can be described via tensor products of elements of ''V''. If, in addition to vector addition and scalar multiplication, there is a bilinear vector product , the vector space is called an Algebra over a field, algebra; for instance, associative algebras are algebras with an associate vector product (like the algebra of square matrices, or the algebra of polynomials).


Topological vector spaces

Vector spaces that are not finite dimensional often require additional structure to be tractable. A normed vector space is a vector space along with a function called a Norm (mathematics), norm, which measures the "size" of elements. The norm induces a Metric (mathematics), metric, which measures the distance between elements, and induces a Topological space, topology, which allows for a definition of continuous maps. The metric also allows for a definition of Limit (mathematics), limits and Complete metric space, completeness - a metric space that is complete is known as a Banach space. A complete metric space along with the additional structure of an Inner product space, inner product (a conjugate symmetric sesquilinear form) is known as a Hilbert space, which is in some sense a particularly well-behaved Banach space. Functional analysis applies the methods of linear algebra alongside those of mathematical analysis to study various function spaces; the central objects of study in functional analysis are Lp space, L''p'' spaces, which are Banach spaces, and especially the ''L''2 space of square integrable functions, which is the only Hilbert space among them. Functional analysis is of particular importance to quantum mechanics, the theory of partial differential equations, digital signal processing, and electrical engineering. It also provides the foundation and theoretical framework that underlies the Fourier transform and related methods.


Homological algebra


See also

* Fundamental matrix (computer vision) * Geometric algebra * Linear programming * Linear regression, a statistical estimation method * List of linear algebra topics * Numerical linear algebra * Transformation matrix


Notes


References


Sources

* * * * * * * * *


Further reading


History

* Fearnley-Sander, Desmond,
Hermann Grassmann and the Creation of Linear Algebra
, American Mathematical Monthly 86 (1979), pp. 809–817. *


Introductory textbooks

* * * * * * * * * Murty, Katta G. (2014)
Computational and Algorithmic Linear Algebra and n-Dimensional Geometry
', World Scientific Publishing, .
Chapter 1: Systems of Simultaneous Linear Equations
' * * * * * The Manga Guide to Linear Algebra (2012), by Shin Takahashi, Iroha Inoue and Trend-Pro Co., Ltd.,


Advanced textbooks

* * * * * * * * * * * * * * * * * * * * * * * * *


Study guides and outlines

* * * * *


External links


Online Resources


MIT Linear Algebra Video Lectures
a series of 34 recorded lectures by Professor Gilbert Strang (Spring 2010)
International Linear Algebra Society
*

on MathWorld
Matrix and Linear Algebra Terms
o



o


Essence of linear algebra
a video presentation from 3Blue1Brown of the basics of linear algebra, with emphasis on the relationship between the geometric, the matrix and the abstract points of view


Online books

* * * * * * * Sharipov, Ruslan,
Course of linear algebra and multidimensional geometry
' * Treil, Sergei,

' {{DEFAULTSORT:Linear Algebra Linear algebra, Numerical analysis