Linear algebra is the branch of

_{1} and ''U''_{2} are subspaces of ''V'', then
:$\backslash dim(U\_1\; +\; U\_2)\; =\; \backslash dim\; U\_1\; +\; \backslash dim\; U\_2\; -\; \backslash dim(U\_1\; \backslash cap\; U\_2),$
where $U\_1+U\_2$ denotes the span of $U\_1\backslash cup\; U\_2.$

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 ...

. Sciences concerned with this space use geometry widely. This is the case with mechanics and robotics, for describing rigid body dynamics;

^{∗} 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).

^{''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.

Hermann Grassmann and the Creation of Linear Algebra

, American Mathematical Monthly 86 (1979), pp. 809–817. *

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.,

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

Course of linear algebra and multidimensional geometry

' * Treil, Sergei,

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

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 ...

s such as:
:$a\_1x\_1+\backslash cdots\; +a\_nx\_n=b,$
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 ...

s such as:
:$(x\_1,\; \backslash ldots,\; x\_n)\; \backslash mapsto\; a\_1x\_1+\backslash cdots\; +a\_nx\_n,$
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 ...

, 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
...

, 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 ...

, 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 ...

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 ...

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 ...

. 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 ...

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 ...

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 ...

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 ...

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 ...

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 $\backslash mathbb$ have a difference ''w'' – ''z'', and the line segments $\backslash overline$ and $\backslash overline$ are of the same length and direction. The segments are equipollent. The four-dimensional system $\backslash mathbb$ 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 ...

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 $\backslash overline\; .$
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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 andmatrices
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 ...

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 ...

'', 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 ...

'', 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 ...

, 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 amap
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 ...

: $T:V\backslash to\; W$
that is compatible with addition and scalar multiplication, that is
: $T(\backslash mathbf\; u\; +\; \backslash mathbf\; v)=T(\backslash mathbf\; u)+T(\backslash mathbf\; v),\; \backslash quad\; T(a\; \backslash mathbf\; v)=aT(\backslash mathbf\; v)$
for any vectors in and scalar in .
This implies that for any vectors in and scalars in , one has
: $T(a\; \backslash mathbf\; u\; +\; b\; \backslash mathbf\; v)=\; T(a\; \backslash mathbf\; u)\; +\; T(b\; \backslash mathbf\; v)\; =\; aT(\backslash mathbf\; u)\; +\; bT(\backslash mathbf\; v)$
When are the same vector space, a linear map $T:V\backslash to\; V$ 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 ...

. 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 ...

.
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 calledlinear 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 ...

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 $T:V\backslash to\; W$, 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 ...

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
: $a\_1\; \backslash mathbf\; v\_1\; +\; a\_2\; \backslash mathbf\; v\_2\; +\; \backslash cdots\; +\; a\_k\; \backslash mathbf\; v\_k,$
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 $S\backslash subseteq\; T,$ then there is a basis such that $S\backslash subseteq\; B\backslash subseteq\; T.$
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 ...

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''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 :$\backslash begin\; (a\_1,\; \backslash ldots,\; a\_m)\&\backslash mapsto\; a\_1\; \backslash mathbf\; v\_1+\backslash cdots\; a\_m\; \backslash mathbf\; v\_m\backslash \backslash \; F^m\; \&\backslash to\; V\; \backslash end$ is abijection
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 ...

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 ...

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 ...

under this isomorphism, that is by the coordinates vector $(a\_1,\; \backslash ldots,\; a\_m)$ 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 ...

:$\backslash begina\_1\backslash \backslash \backslash vdots\backslash \backslash a\_m\backslash end.$
If is another finite dimensional vector space (possibly the same), with a basis $(\backslash mathbf\; w\_1,\; \backslash ldots,\; \backslash mathbf\; w\_n),$ a linear map from to is well defined by its values on the basis elements, that is $(f(\backslash mathbf\; w\_1),\; \backslash ldots,\; f(\backslash mathbf\; w\_n)).$ Thus, is well represented by the list of the corresponding column matrices. That is, if
:$f(w\_j)=a\_v\_1\; +\; \backslash cdots+a\_v\_m,$
for , then is represented by the matrix
:$\backslash begin\; a\_\&\backslash cdots\&a\_\backslash \backslash \; \backslash vdots\&\backslash ddots\&\backslash vdots\backslash \backslash \; a\_\&\backslash cdots\&a\_\; \backslash end,$
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 ...

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 ...

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, $x\_1,\; x\_2,\; \backslash ldots,\; x\_n$ or $x,\; y,\; \backslash ldots,\; z$ 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 :$M\; =\; \backslash left;\; href="/html/ALL/s/begin\; 2\_\_1\_\_-1\backslash \backslash \; -3\_\_-1\_\_2\_\_\backslash \backslash \; -2\_\_1\_\_2\; \backslash end\backslash right.html"\; ;"title="begin\; 2\; 1\; -1\backslash \backslash \; -3\; -1\; 2\; \backslash \backslash \; -2\; 1\; 2\; \backslash end\backslash right">begin\; 2\; 1\; -1\backslash \backslash \; -3\; -1\; 2\; \backslash \backslash \; -2\; 1\; 2\; \backslash end\backslash right$ and its right member vector :$\backslash mathbf\; =\; \backslash begin\; 8\backslash \backslash -11\backslash \backslash -3\; \backslash end.$ Let be the linear transformation associated to the matrix . A solution of the system () is a vector :$\backslash mathbf=\backslash begin\; x\backslash \backslash y\backslash \backslash z\; \backslash end$ such that :$T(\backslash mathbf)\; =\; \backslash mathbf,$ that is an element of thepreimage
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 augmented matrix
Augment or augmentation may refer to:
Language
* Augment (Indo-European), a syllable added to the beginning of the word in certain Indo-European languages
* Augment (Bantu languages), a morpheme that is prefixed to the noun class prefix of nouns ...

:$\backslash left;\; href="/html/ALL/s/!\backslash beginM\backslash mathbf\backslash end\backslash !\backslash right.html"\; ;"title="!\backslash beginM\backslash mathbf\backslash end\backslash !\backslash right">!\backslash beginM\backslash mathbf\backslash end\backslash !\backslash right$
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
:$\backslash left;\; href="/html/ALL/s/!\backslash beginM\backslash mathbf\backslash end\backslash !\backslash right.html"\; ;"title="!\backslash beginM\backslash mathbf\backslash end\backslash !\backslash right">!\backslash beginM\backslash mathbf\backslash end\backslash !\backslash right$
showing that the system () has the unique solution
:$\backslash beginx\&=2\backslash \backslash y\&=3\backslash \backslash z\&=-1.\backslash end$
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, matrix inverseIn 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 ...

s.
Endomorphisms and square matrices

A linearendomorphism
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 :$\backslash sum\_\; (-1)^\; a\_\; \backslash cdots\; a\_,$ where $S\_n$ is the group of all permutations of elements, $\backslash sigma$ is a permutation, and $(-1)^\backslash sigma$ 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 theidentity 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 ...

, 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 $\backslash det(M-aI)$ equals zero. The eigenvalues are thus the roots
A root
In vascular plant
Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a lar ...

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 ...

:$\backslash det(xI-M).$
If is of dimension , this is a monic polynomial
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...

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 square-free{{no footnotes, date=December 2015
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 (math ...

, 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
:$\backslash begin0\&1\backslash \backslash 0\&0\backslash end$
(it cannot be diagonalizable since its square is the zero matrixIn 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 ...

, 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

Alinear 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 ...

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 $V\text{'}$.
If $\backslash mathbf\; v\_1,\; \backslash ldots,\; \backslash mathbf\; v\_n$ is a basis of $V$ (this implies that is finite-dimensional), then one can define, for , a linear map $v\_i^*$ such that $v\_i^*(\backslash mathbf\; e\_i)=1$ and $v\_i^*(\backslash mathbf\; e\_j)=0$ 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 $\backslash mathbf\; v\_1,\; \backslash ldots,\; \backslash mathbf\; v\_n.$ (If is not finite-dimensional, the $v^*\_i$ may be defined similarly; they are linearly independent, but do not form a basis.)
For $\backslash mathbf\; v$ in $V$, the map
:$f\backslash to\; f(\backslash mathbf\; v)$
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 ...

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 ...

:$\backslash langle\; f,\; \backslash mathbf\; x\backslash rangle$
for denoting $f(\backslash mathbf\; x)$.
Dual map

Let :$f:V\backslash to\; W$ be a linear map. For every linear form on , the composite function is a linear form on . This defines a linear map :$f^*:W^*\backslash to\; V^*$ 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 thetranspose
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 ...

$M^\backslash mathsf\; T$ 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
:$\backslash langle\; h^\backslash mathsf\; T\; ,\; M\; \backslash mathbf\; v\backslash rangle\; =\; \backslash langle\; h^\backslash mathsf\; T\; M,\; \backslash mathbf\; v\backslash rangle.$
For highlighting this symmetry, the two members of this equality are sometimes written
:$\backslash langle\; h^\backslash mathsf\; T\; \backslash mid\; M\; \backslash mid\; \backslash mathbf\; v\backslash rangle.$
Inner-product spaces

Besides these basic concepts, linear algebra also studies vector spaces with additional structure, such as aninner 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
:$\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle\; :\; V\; \backslash times\; V\; \backslash to\; F$
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 ...

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:
*::$\backslash langle\; \backslash mathbf\; u,\; \backslash mathbf\; v\backslash rangle\; =\backslash overline.$
*: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 ...

ity in the first argument:
*::$\backslash langle\; a\; \backslash mathbf\; u,\; \backslash mathbf\; v\backslash rangle=\; a\; \backslash langle\; \backslash mathbf\; u,\; \backslash mathbf\; v\backslash rangle.$
*::$\backslash langle\; \backslash mathbf\; u\; +\; \backslash mathbf\; v,\; \backslash mathbf\; w\backslash rangle=\; \backslash langle\; \backslash mathbf\; u,\; \backslash mathbf\; w\backslash rangle+\; \backslash langle\; \backslash mathbf\; v,\; \backslash mathbf\; w\backslash rangle.$
* Definite bilinear form, Positive-definiteness:
*::$\backslash langle\; \backslash mathbf\; v,\; \backslash mathbf\; v\backslash rangle\; \backslash geq\; 0$ with equality only for v = 0.
We can define the length of a vector v in ''V'' by
:$\backslash ,\; \backslash mathbf\; v\backslash ,\; ^2=\backslash langle\; \backslash mathbf\; v,\; \backslash mathbf\; v\backslash rangle,$
and we can prove the Cauchy–Schwarz inequality:
:$,\; \backslash langle\; \backslash mathbf\; u,\; \backslash mathbf\; v\backslash rangle,\; \backslash leq\; \backslash ,\; \backslash mathbf\; u\backslash ,\; \backslash cdot\; \backslash ,\; \backslash mathbf\; v\backslash ,\; .$
In particular, the quantity
:$\backslash frac\; \backslash leq\; 1,$
and so we can call this quantity the cosine of the angle between the two vectors.
Two vectors are orthogonal if $\backslash langle\; \backslash mathbf\; u,\; \backslash mathbf\; v\backslash rangle\; =0$. 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 $a\_i\; =\; \backslash langle\; \backslash mathbf\; v,\; \backslash mathbf\; v\_i\; \backslash rangle$.
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
:$\backslash langle\; T\; \backslash mathbf\; u,\; \backslash mathbf\; v\; \backslash rangle\; =\; \backslash langle\; \backslash mathbf\; u,\; T^*\; \backslash mathbf\; v\backslash rangle.$
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 andgeometry
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 ...

, 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 ...

, 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
...

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 ...

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 ongeodesy
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 ...

. 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 thatdeterminant
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 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''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, LHomological 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 matrixNotes

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

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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