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Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.^{[1]}
Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinitedimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a welldeveloped theory, nonlinear mathematical models are sometimes approximated by linear ones.
The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. Determinants were used by Leibniz in 1693, and subsequently, Gabriel Cramer devised Cramer's Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, which was initially listed as an advancement in geodesy.^{[2]}
The study of matrix algebra first emerged in England in the mid1800s. In 1844 Hermann Grassmann published his “Theory of Extension” which included foundational new topics of what is today called linear algebra. In 1848, James Joseph Sylvester introduced the term matrix, which is Latin for "womb". While studying compositions of linear transformations, Arthur Cayley was led to define matrix multiplication and inverses. 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".^{[3]}
In 1882, Hüseyin Tevfik Pasha wrote the book titled "Linear Algebra".^{[4]}^{[5]} The first modern and more precise definition of a vector space was introduced by Peano in 1888;^{[3]} by 1900, a theory of linear transformations of finitedimensional vector spaces had emerged. Linear algebra first 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. The use of matrices in quantum mechanics, special relativity, and statistics helped spread the subject of linear algebra beyond pure mathematics. The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations.^{[3]}
The origin of many of these ideas is discussed in the articles on determinants and Gaussian elimination.
The main structures of linear algebra are vector spaces. A vector space over a field F is a set V together with two binary operations. Elements of V are called vectors and elements of F are called scalars. The first operation, vector addition, takes any two vectors v and w and outputs a third vector v + w. The second operation takes any scalar a and any vector v and outputs a new vector vector av. In view of the first example, where the multiplication is done by rescaling the vector v by a scalar a, the multiplication is called scalar multiplication of v by a. The operations of addition and multiplication in a vector space satisfy the following axioms.^{[6]} In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars in F.
Axiom  Signification 
Associativity of addition  u + (v + w) = (u + v) + w 
Commutativity of addition  u + v = v + u 
Identity element of addition  There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V. 
Inverse elements of addition  For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0 
Distributivity of scalar multiplication with respect to vector addition  a(u + v) = au + av 
Distributivity of scalar multiplication with respect to field addition  (a + b)v = av + bv 
Compatibility of scalar multiplication with field multiplication  a(bv) = (ab)v ^{[nb 1]} 
Identity element of scalar multiplication  1v = v, where 1 denotes the multiplicative identity in F. 
Elements of a general vector space V may be objects of any nature, for example, functions, polynomials, vectors, or matrices. Linear algebra is concerned with properties common to all vector spaces.
Similarly as in the theory of other algebraic structures, linear algebra studies mappings between vector spaces that preserve the vectorspace structure. Given two vector spaces V and W over a field F, a linear transformation (also called linear map, linear mapping or linear operator) is a map
that is compatible with addition and scalar multiplication:
for any vectors u,v ∈ V and a scalar a ∈ F.
Additionally for any vectors u, v ∈ V and scalars a, b ∈ F:
When a bijective linear mapping exists between two vector spaces (that is, every vector from the second space is associated with exactly one in the first), we say that the two spaces are isomorphic. Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view. One essential question in linear algebra is whether a mapping is an isomorphism or not, and this question can be answered by checking if the determinant is nonzero. If a mapping is not an isomorphism, linear algebra is interested in finding its range (or image) and the set of elements that get mapped to zero, called the kernel of the mapping.
Linear transformations have geometric significance. For example, 2 × 2 real matrices denote standard planar mappings that preserve the origin.
Again in analogue with theories of other algebraic objects, linear algebra is interested in subsets of vector spaces that are vector spaces themselves; these subsets are called linear subspaces. For instance, the range and kernel of a linear mapping are both subspaces, and are thus often called the range space and the nullspace; these are important examples of subspaces. Another important way of forming a subspace is to take a linear combination of a set of vectors v_{1}, v_{2}, …, v_{k}:
where a_{1}, a_{2}, …, a_{k} are scalars. The set of all linear combinations of vectors v_{1}, v_{2}, …, v_{k} is called their span, which forms a subspace.
A linear combination of any system of vectors with all zero coefficients is the zero vector of V. If this is the only way to express zero vector as a linear combination of v_{1}, v_{2}, …, v_{k} then these vectors are linearly independent. Given a set of vectors that span a space, if any vector w is a linear combination of other vectors (and so the set is not linearly independent), then the span would remain the same if we remove w from the set. Thus, a set of linearly dependent vectors is redundant in the sense that a linearly independent subset will span the same subspace. Therefore, we are mostly interested in a linearly independent set of vectors that spans a vector space V, which we call a basis of V. Any set of vectors that spans V contains a basis, and any linearly independent set of vectors in V can be extended to a basis.^{[7]} It turns out that if we accept the axiom of choice, every vector space has a basis;^{[8]} nevertheless, this basis may be unnatural, and indeed, may not even be constructable. For instance, there exists a basis for the real numbers considered as a vector space over the rationals, but no explicit basis has been constructed.
Any two bases of a vector space V have the same cardinality, which is called the dimension of V. The dimension of a vector space is welldefined by the dimension theorem for vector spaces. If a basis of V has finite number of elements, V is called a finitedimensional vector space. If V is finitedimensional and U is a subspace of V, then dim U ≤ dim V. If U_{1} and U_{2} are subspaces of V, then
One often restricts consideration to finitedimensional vector spaces. A fundamental theorem of linear algebra states that all vector spaces of the same dimension are isomorphic,^{[10]} giving an easy way of characterizing isomorphism.
A particular basis {v_{1}, v_{2}, …, v_{n}} of V allows one to construct a coordinate system in V: the vector with coordinates (a_{1}, a_{2}, …, a_{n}) is the linear combination
The condition that v_{1}, v_{2}, …, v_{n} span V guarantees that each vector v can be assigned coordinates, whereas the linear independence of v_{1}, v_{2}, …, v_{n} assures that these coordinates are unique (i.e. there is only one linear combination of the basis vectors that is equal to v). In this way, once a basis of a vector space V over F has been chosen, V may be identified with the coordinate nspace F^{n}. Under this identification, addition and scalar multiplication of vectors in V correspond to addition and scalar multiplication of their coordinate vectors in F^{n}. Furthermore, if V and W are an ndimensional and mdimensional vector space over F, and a basis of V and a basis of W have been fixed, then any linear transformation T: V → W may be encoded by an m × n matrix A with entries in the field F, called the matrix of T with respect to these bases. Two matrices that encode the same linear transformation in different bases are called similar. Matrix theory replaces the study of linear transformations, which were defined axiomatically, by the study of matrices, which are concrete objects. This major technique distinguishes linear algebra from theories of other algebraic structures, which usually cannot be parameterized so concretely.
There is an important distinction between the coordinate nspace R^{n} and a general finitedimensional vector space V. While R^{n} has a standard basis {e_{1}, e_{2}, …, e_{n}}, a vector space V typically does not come equipped with such a basis and many different bases exist (although they all consist of the same number of elements equal to the dimension of V).
One major application of the matrix theory is calculation of determinants, a central concept in linear algebra. While determinants could be defined in a basisfree manner, they are usually introduced via a specific representation of the mapping; the value of the determinant does not depend on the specific basis. It turns out that a mapping is invertible if and only if the determinant is nonzero. If the determinant is zero, then the nullspace is nontrivial. Determinants have other applications, including a systematic way of seeing if a set of vectors is linearly independent (we write the vectors as the columns of a matrix, and if the determinant of that matrix is zero, the vectors are linearly dependent). Determinants could also be used to solve systems of linear equations (see Cramer's rule), but in real applications, Gaussian elimination is a faster method.
In general, the action of a linear transformation may be quite complex. Attention to lowdimensional examples gives an indication of the variety of their types. One strategy for a general ndimensional transformation T is to find "characteristic lines" that are invariant sets under T. If v is a nonzero vector such that Tv is a scalar multiple of v, then the line through 0 and v is an invariant set under T and v is called a characteristic vector or eigenvector. The scalar λ such that Tv = λv is called a characteristic value or eigenvalue of T.
To find an eigenvector or an eigenvalue, we note that
where I is the identity matrix. For there to be nontrivial solutions to that equation, det(T − λ I) = 0. The determinant is a polynomial, and so the eigenvalues are not guaranteed to exist if the field is R. Thus, we often work with an algebraically closed field such as the complex numbers when dealing with eigenvectors and eigenvalues so that an eigenvalue will always exist. It would be particularly nice if given a transformation T taking a vector space V into itself we can find a basis for V consisting of eigenvectors. If such a basis exists, we can easily compute the action of the transformation on any vector: if v_{1}, v_{2}, …, v_{n} are linearly independent eigenvectors of a mapping of ndimensional spaces T with (not necessarily distinct) eigenvalues λ_{1}, λ_{2}, …, λ_{n}, and if v = a_{1}v_{1} + ... + a_{n} v_{n}, then,
Such a transformation is called a diagonalizable matrix since in the eigenbasis, the transformation is represented by a diagonal matrix. Because operations like matrix multiplication, matrix inversion, and determinant calculation are simple on diagonal matrices, computations involving matrices are much simpler if we can bring the matrix to a diagonal form. Not all matrices are diagonalizable (even over an algebraically closed field).
Besides these basic concepts, linear algebra also studies vector spaces with additional structure, such as an inner product. The inner product is an example of a bilinear form, 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
that satisfies the following three axioms for all vectors u, v, w in V and all scalars a in F:^{[11]}^{[12]}
Note that in R, it is symmetric.
We can define the length of a vector v in V by
and we can prove the Cauchy–Schwarz inequality:
In particular, the quantity
and so we can call this quantity the cosine of the angle between the two vectors.
Two vectors are orthogonal if . An orthonormal basis is a basis where all basis vectors have length 1 and are orthogonal to each other. Given any finitedimensional vector space, an orthonormal basis could be found by the Gram–Schmidt procedure. Orthonormal bases are particularly nice to deal with, since if v = a_{1} v_{1} + ... + a_{n} v_{n}, then .
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
If T satisfies TT* = T*T, we call T normal. It turns out that normal matrices are precisely the matrices that have an orthonormal system of eigenvectors that span V.
Because of the ubiquity of vector spaces, linear algebra is used in many fields of mathematics, natural sciences, computer science, and social science. Below are just some examples of applications of linear algebra.
Linear algebra provides the formal setting for the linear combination of equations used in the Gaussian method. Suppose the goal is to find and describe the solution(s), if any, of the following system of linear equations:
The Gaussianelimination algorithm is as follows: eliminate x from all equations below L_{1}, and then eliminate y from all equations below L_{2}. This will put the system into triangular form. Then, using backsubstitution, each unknown can be solved for.
In the example, x is eliminated from L_{2} by adding (3/2)L_{1} to L_{2}. x is then eliminated from L_{3} by adding L_{1} to L_{3}. Formally:
The result is:
Now y is eliminated from L_{3} by adding −4L_{2} to L_{3}:
The result is:
This result is a system of linear equations in triangular form, and so the first part of the algorithm is complete.
The last part, backsubstitution, consists of solving for the knowns in reverse order. It can thus be seen that
Then, z can be substituted into L_{2}, which can then be solved to obtain
Next, z and y can be substituted into L_{1}, which can be solved to obtain
The system is solved.
We can, in general, write any system of linear equations as a matrix equation:
The solution of this system is characterized as follows: first, we find a particular solution x_{0} of this equation using Gaussian elimination. Then, we compute the solutions of Ax = 0; that is, we find the nullspace N of A. The solution set of this equation is given by . If the number of variables equal the number of equations, then we can characterize when the system has a unique solution: since N is trivial if and only if det A ≠ 0, the equation has a unique solution if and only if det A ≠ 0.^{[13]}
The least squares method is used to determine the best fit line for a set of data.^{[14]} This line will minimize the sum of the squares of the residuals.
Fourier series are a representation of a function f: [−π, π] → R as a trigonometric series:
This series expansion is extremely useful in solving partial differential equations. In this article, we will not be concerned with convergence issues; it is nice to note that all Lipschitzcontinuous functions have a converging Fourier series expansion, and nice enough discontinuous functions have a Fourier series that converges to the function value at most points.
The space of all functions that can be represented by a Fourier series form a vector space (technically speaking, we call functions that have the same Fourier series expansion the "same" function, since two different discontinuous functions might have the same Fourier series). Moreover, this space is also an inner product space with the inner product
The functions g_{n}(x) = sin(nx) for n > 0 and h_{n}(x) = cos(nx) for n ≥ 0 are an orthonormal basis for the space of Fourierexpandable functions. We can thus use the tools of linear algebra to find the expansion of any function in this space in terms of these basis functions. For instance, to find the coefficient a_{k}, we take the inner product with h_{k}:
and by orthonormality, ; that is,
Quantum mechanics is highly inspired by notions in linear algebra. In quantum mechanics, the physical state of a particle is represented by a vector, and observables (such as momentum, energy, and angular momentum) are represented by linear operators on the underlying vector space. More concretely, the wave function of a particle describes its physical state and lies in the vector space L^{2} (the functions φ: R^{3} → C such that is finite), and it evolves according to the Schrödinger equation. Energy is represented as the operator , where V is the potential energy. H is also known as the Hamiltonian operator. The eigenvalues of H represents the possible energies that can be observed. Given a particle in some state φ, we can expand φ into a linear combination of eigenstates of H. The component of H in each eigenstate determines the probability of measuring the corresponding eigenvalue, and the measurement forces the particle to assume that eigenstate (wave function collapse).
Since linear algebra is a successful theory, its methods have been developed and generalized in other parts of mathematics. In module theory, one replaces the field of scalars by a ring. The concepts of linear independence, span, basis, and dimension (which is called rank in module theory) still make sense. Nevertheless, many theorems from linear algebra become false in module theory. For instance, not all modules have a basis (those that do are called free modules), the rank of a free module is not necessarily unique, not all linearly independent subsets of a module can be extended to form a basis, and not all subsets of a module that span the space contains a basis.
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 f: V → F where F is the field of scalars. Multilinear maps T: V^{n} → F 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, then the vector space is called an algebra; for instance, associative algebras are algebras with an associate vector product (like the algebra of square matrices, or the algebra of polynomials).
Functional analysis mixes the methods of linear algebra with those of mathematical analysis and studies various function spaces, such as Lp spaces.
Representation theory studies the actions of algebraic objects on vector spaces by representing these objects as matrices. It is interested in all the ways that this is possible, and it does so by finding subspaces invariant under all transformations of the algebra. The concept of eigenvalues and eigenvectors is especially important.
Algebraic geometry considers the solutions of systems of polynomial equations.
Wikibooks has a book on the topic of: Linear Algebra 
