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The three-dimensional Euclidean space R³ is a vector space, and lines and planes passing through the origin are vector subspaces in R³.

Linear algebra is the branch of mathematics concerning finite or countably infinite dimensional vector spaces, as well as linear mappings between such spaces. Such an investigation is initially motivated by a system of linear equations in 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 leading to a number of generalizations. Functional analysis studies the infinite-dimensional 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, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear ones.

History[link]

The study of linear algebra and matrices 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, Cramer devised the 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 mid 1800s. In 1848, Sylvester introduced the term matrix, which is Latin for "womb". While studying compositions linear transformations, Arthur Cayley was led to define matrix multiplication and inverses. Crucially, Cayley used a single letter to denote a matrix, thus thinking of matrices as an aggregate object. He also realized the connection between matrices and determinants and wrote that "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants".^[3]

The first modern and more precise definition of a vector space was introduced by Peano in 1888;^[4] by 1900, a theory of linear transformations of finite-dimensional 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.^[5]

The origin of many of these ideas is discussed in the articles on determinants and Gaussian elimination.

Scope of study[link]

Vector spaces[link]

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 satisfies following axioms.^[6] In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars 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.

Linear transformations[link]

Similarly as in the theory of other algebraic structures, linear algebra studies mappings between vector spaces that preserve the vector-space 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

Failed to parse (Missing texvc executable; please see math/README to configure.): T:V\to W

that is compatible with addition and scalar multiplication:

Failed to parse (Missing texvc executable; please see math/README to configure.): T(u+v)=T(u)+T(v), \quad T(av)=aT(v)

for any vectors u,v ∈ V and a scalar a ∈ F. When a bijective linear mapping exists between two vector spaces (that is, every vector from the first space is associated with one in the second), 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.

Subspaces, span, and basis[link]

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 v₁, v₂, …, v_k:

Failed to parse (Missing texvc executable; please see math/README to configure.): a_1 v_1 + a_2 v_2 + \cdots + a_k v_k, \,

where a₁, a₂, …, a_k are scalars. The set of all linear combinations of vectors v₁, v₂, …, 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 v₁, v₂, …, v_k then these vectors are linearly independent. Given a set of vectors that span a space, if any vector Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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 well-defined by the dimension theorem for vector spaces. If a basis of V has finite number of elements, V is called a finite-dimensional vector space. If V is finite-dimensional and U is a subspace of V, then dim U ≤ dim V. If U₁ and U₂ are subspaces of V, then

Failed to parse (Missing texvc executable; please see math/README to configure.): \dim(U_1 + U_2) = \dim U_1 + \dim U_2 - \dim(U_1 \cap U_2)

.^[9] One often restricts consideration to finite-dimensional 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.

[edit] Vectors as n-tuples: matrix theory

A particular basis {v₁, v₂, …, v_n} of V allows one to construct a coordinate system in V: the vector with coordinates (a₁, a₂, …, a_n) is the linear combination

Failed to parse (Missing texvc executable; please see math/README to configure.): a_1 v_1 + a_2 v_2 + \cdots + a_n v_n. \,

The condition that v₁, v₂, …, v_n span V guarantees that each vector v can be assigned coordinates, whereas the linear independence of v₁, v₂, …, 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 n-space Fⁿ. Under this identification, addition and scalar multiplication of vectors in V correspond to addition and scalar multiplication of their coordinate vectors in Fⁿ. Furthermore, if V and W are an n-dimensional and m-dimensional 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 parametrized so concretely.

There is an important distinction between the coordinate n-space Rⁿ and a general finite-dimensional vector space V. While Rⁿ has a standard basis {e₁, e₂, …, e_n}, a vector space V typically does not come equipped with 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 basis-free 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.

Eigenvalues and eigenvectors[link]

In general, the action of a linear transformation is hard to understand, and so to get a better handle over linear transformations, those vectors that are relatively fixed by that transformation are given special attention. To make this more concrete, let Failed to parse (Missing texvc executable; please see math/README to configure.): T: V \to V

be any linear transformation. We are especially interested in those non-zero vectors Failed to parse (Missing texvc executable; please see math/README to configure.): v
such that Failed to parse (Missing texvc executable; please see math/README to configure.): Tv=\lambda v

, where Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda

is a scalar in the base field of the vector space. These vectors are called eigenvectors, and the corresponding scalars are called eigenvalues. 

To find an eigenvector or an eigenvalue, we note that

Failed to parse (Missing texvc executable; please see math/README to configure.): Tv-\lambda v=(T-\lambda \text{Id})v=0,

where Failed to parse (Missing texvc executable; please see math/README to configure.): \text{Id}

is the identity matrix. 

For there to be nontrivial solutions to that equation, Failed to parse (Missing texvc executable; please see math/README to configure.): \det(T-\lambda \text{Id})=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 Failed to parse (Missing texvc executable; please see math/README to configure.): T

taking a vector space Failed to parse (Missing texvc executable; please see math/README to configure.): V
into itself we can find a basis for Failed to parse (Missing texvc executable; please see math/README to configure.): V
consisting of eigenvectors. If such a basis exists, we can easily compute the action of the transformation on any vector:

if Failed to parse (Missing texvc executable; please see math/README to configure.): v_1, v_2, \ldots, v_n

are linearly independent eigenvectors of a mapping of n-dimensional spaces Failed to parse (Missing texvc executable; please see math/README to configure.): T
 with (not necessarily distinct) eigenvalues Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda_1, \lambda_2, \ldots, \lambda_n

, and if Failed to parse (Missing texvc executable; please see math/README to configure.): v=a_1 v_1 + \cdots + a_n v_n , then,

Failed to parse (Missing texvc executable; please see math/README to configure.): T(v)=T(a_1 v_1)+\cdots+T(a_n v_n)=a_1 T(v_1)+\cdots+a_n T(v_n)=a_1 \lambda_1 v_1 + \cdots +a_n \lambda_n v_n.

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), but diagonalizable matrices form a dense subset of all matrices.

Inner-product spaces[link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle \cdot, \cdot \rangle : V \times V \rightarrow \mathbf{F}

that satisfies the following three axioms for all vectors Failed to parse (Missing texvc executable; please see math/README to configure.): u,v,w \in V

and all scalars Failed to parse (Missing texvc executable; please see math/README to configure.): a \in \mathbf{F}

^[11][12]

• Conjugate symmetry:

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle u,v\rangle =\overline{\langle v,u\rangle}.

Note that in R, it is symmetric.

• Linearity in the first argument:

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle au,v\rangle= a \langle u,v\rangle.

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle u+v,w\rangle= \langle u,w\rangle+ \langle v,w\rangle.

• Positive-definiteness:

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle v,v\rangle \geq 0

with equality only for Failed to parse (Missing texvc executable; please see math/README to configure.): v = 0.

We can define the length of a vector Failed to parse (Missing texvc executable; please see math/README to configure.): v \in V

by Failed to parse (Missing texvc executable; please see math/README to configure.): ||v||^2=\langle v,v\rangle

, and we can prove the Cauchy–Schwartz inequality:

Failed to parse (Missing texvc executable; please see math/README to configure.): |\langle u,v\rangle| \leq ||u|| \cdot ||v||.

In particular, the quantity

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{|\langle u,v\rangle|}{||u|| \cdot ||v||} \leq 1,

and so we can call this quantity the cosine of the angle between the two vectors.

Two vectors are orthogonal if Failed to parse (Missing texvc executable; please see math/README to configure.): \langle u, 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 nice to deal with, since if Failed to parse (Missing texvc executable; please see math/README to configure.): v=a_1 v_1 + \cdots + a_n v_n , then Failed to parse (Missing texvc executable; please see math/README to configure.): a_i = \langle v,v_i \rangle .

The inner product facilitates the construction of many useful concepts. For instance, given a transform Failed to parse (Missing texvc executable; please see math/README to configure.): T , we can define its Hermitian conjugate Failed to parse (Missing texvc executable; please see math/README to configure.): T^*

as the linear transform satisfying

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle T u, v \rangle = \langle u, T^* v\rangle.

If T satisfies Failed to parse (Missing texvc executable; please see math/README to configure.): T T^*=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.

Some main useful theorems[link]

• A matrix is invertible, or non-singular, if and only if the linear map represented by the matrix is an isomorphism.
• Any vector space over a field F of dimension n is isomorphic to Fⁿ as a vector space over F.
• Corollary: Any two vector spaces over F of the same finite dimension are isomorphic to each other.
• A linear map is an isomorphism if and only if the determinant is nonzero.

Applications[link]

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.

Solution of linear systems[link]

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{alignat}{7} 2x &&\; + \;&& y &&\; - \;&& z &&\; = \;&& 8 & \qquad (L_1) \\ -3x &&\; - \;&& y &&\; + \;&& 2z &&\; = \;&& -11 & \qquad (L_2) \\ -2x &&\; + \;&& y &&\; +\;&& 2z &&\; = \;&& -3 & \qquad (L_3) \end{alignat}

The Gaussian-elimination algorithm is as follows: eliminate x from all equations below Failed to parse (Missing texvc executable; please see math/README to configure.): L_1 , and then eliminate y from all equations below Failed to parse (Missing texvc executable; please see math/README to configure.): L_2 . This will put the system into triangular form. Then, using back-substitution, each unknown can be solved for.

In the example, x is eliminated from Failed to parse (Missing texvc executable; please see math/README to configure.): L_2

by adding Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{matrix}\frac{3}{2}\end{matrix} L_1
to Failed to parse (Missing texvc executable; please see math/README to configure.): L_2

. x is then eliminated from Failed to parse (Missing texvc executable; please see math/README to configure.): L_3

by adding Failed to parse (Missing texvc executable; please see math/README to configure.): L_1
to Failed to parse (Missing texvc executable; please see math/README to configure.): L_3

. Formally:

Failed to parse (Missing texvc executable; please see math/README to configure.): L_2 + \frac{3}{2}L_1 \rightarrow L_2

Failed to parse (Missing texvc executable; please see math/README to configure.): L_3 + L_1 \rightarrow L_3

The result is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{alignat}{7} 2x &&\; + && y &&\; - &&\; z &&\; = \;&& 8 & \\ && && \frac{1}{2}y &&\; + &&\; \frac{1}{2}z &&\; = \;&& 1 & \\ && && 2y &&\; + &&\; z &&\; = \;&& 5 & \end{alignat}

Now y is eliminated from Failed to parse (Missing texvc executable; please see math/README to configure.): L_3

by adding Failed to parse (Missing texvc executable; please see math/README to configure.): -4L_2
to Failed to parse (Missing texvc executable; please see math/README to configure.): L_3

Failed to parse (Missing texvc executable; please see math/README to configure.): L_3 + -4L_2 \rightarrow L_3

The result is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{alignat}{7} 2x &&\; + && y \;&& - &&\; z \;&& = \;&& 8 & \\ && && \frac{1}{2}y \;&& + &&\; \frac{1}{2}z \;&& = \;&& 1 & \\ && && && &&\; -z \;&&\; = \;&& 1 & \end{alignat}

This result is a system of linear equations in triangular form, and so the first part of the algorithm is complete.

The last part, back-substitution, consists of solving for the knowns in reverse order. It can thus be seen that

Failed to parse (Missing texvc executable; please see math/README to configure.): z = -1 \quad (L_3)

Then, Failed to parse (Missing texvc executable; please see math/README to configure.): z

can be substituted into Failed to parse (Missing texvc executable; please see math/README to configure.): L_2

, which can then be solved to obtain

Failed to parse (Missing texvc executable; please see math/README to configure.): y = 3 \quad (L_2)

Next, z and y can be substituted into Failed to parse (Missing texvc executable; please see math/README to configure.): L_1 , which can be solved to obtain

Failed to parse (Missing texvc executable; please see math/README to configure.): x = 2 \quad (L_1)

The system is solved.

We can, in general, write any system of linear equations as a matrix equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): Ax=b.

The solution of this system is characterized as follows: first, we find a particular solution Failed to parse (Missing texvc executable; please see math/README to configure.): x_0

of this equation using Gaussian elimination.

Then, we compute the solutions of Failed to parse (Missing texvc executable; please see math/README to configure.): Ax=0

that is, we find the nullspace Failed to parse (Missing texvc executable; please see math/README to configure.): N

of A.

The solution set of this equation is given by Failed to parse (Missing texvc executable; please see math/README to configure.): x_0+N=\{x_0+n: n\in N \} . 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \det A \neq 0 , the equation has a unique solution if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): \det A \neq 0 . ^[13]

Least-squares best fit line[link]

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 expansion[link]

Fourier series are a representation of a function Failed to parse (Missing texvc executable; please see math/README to configure.): f:[-\pi,\pi] \to \mathbf{R}

as a trigonometric series:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=\frac{a_0}{2} + \sum_{n=1}^\infty \, [a_n \cos(nx) + b_n \sin(nx)].

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 continuous 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle f,g \rangle= \frac{1}{\pi} \int_{-\pi}^\pi f(x) g(x) \, dx.

The functions Failed to parse (Missing texvc executable; please see math/README to configure.): g_n(x)=\sin(nx)

for Failed to parse (Missing texvc executable; please see math/README to configure.): n>0
and Failed to parse (Missing texvc executable; please see math/README to configure.): h_n(x)=\cos(nx)
for Failed to parse (Missing texvc executable; please see math/README to configure.): n \geq 0
are an orthonormal basis for the space of Fourier-expandable 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 Failed to parse (Missing texvc executable; please see math/README to configure.): a_k , we take the inner product with Failed to parse (Missing texvc executable; please see math/README to configure.): h_k

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle f,h_k \rangle=\frac{a_0}{2}\langle h_0,h_k \rangle + \sum_{n=1}^\infty \, [a_n \langle h_n,h_k\rangle + b_n \langle\ g_n,h_k \rangle],

and by orthonormality, Failed to parse (Missing texvc executable; please see math/README to configure.): \langle f,h_k\rangle=a_k

that is, Failed to parse (Missing texvc executable; please see math/README to configure.): a_k = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \cos(kx) \, dx.

Quantum mechanics[link]

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 L² (the functions Failed to parse (Missing texvc executable; please see math/README to configure.): \phi:\mathbf{R}^3 \to \mathbf{C}

such that Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^{\infty} |\phi|^2 \,dx\,dy\,dz
is finite), and it evolves according to the Schrödinger equation. 

Energy is represented as the operator Failed to parse (Missing texvc executable; please see math/README to configure.): H=-\frac{\hbar^2}{2m} \nabla^2 + V(x,y,z) , 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \phi , we can expand Failed to parse (Missing texvc executable; please see math/README to configure.): \phi

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).

Generalizations and related topics[link]

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 Failed to parse (Missing texvc executable; please see math/README to configure.): V^*

consisting of linear maps Failed to parse (Missing texvc executable; please see math/README to configure.): f:V \to \mathbf{F}
where F is the field of scalars. Multilinear maps Failed to parse (Missing texvc executable; please see math/README to configure.): T: V^n \to \mathbf{F}
can be described via tensor products of elements of Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

See also[link]

• List of linear algebra topics
• Numerical linear algebra
• Eigenvectors
• Transformation matrix
• Fundamental matrix in computer vision
• Simplex method, a solution technique for linear programs
• Linear regression, a statistical estimation method

Notes[link]

1. ^ Weisstein, Eric. "Linear Algebra". From MathWorld--A Wolfram Web Resource.. Wolfram. http://mathworld.wolfram.com/LinearAlgebra.html. Retrieved 16 April 2012. 
2. ^ Vitulli, Marie. "A Brief History of Linear Algebra and Matrix Theory". Department of Mathematics. University of Oregon. http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html. Retrieved 01/24/2012. 
3. ^ Vitulli, Marie
4. ^ Vitulli, Marie
5. ^ Vitulli, Marie
6. ^ Roman 2005, ch. 1, p. 27
7. ^ Axler (2004), pp. 28–29
8. ^ The existence of a basis is straightforward for countably generated vector spaces, and for well-ordered vector spaces, but in full generality it is logically equivalent to the axiom of choice.
9. ^ Axler (2204), p. 33
10. ^ Axler (2004), p. 55
11. ^ P. K. Jain, Khalil Ahmad (1995). "5.1 Definitions and basic properties of inner product spaces and Hilbert spaces". Functional analysis (2nd ed.). New Age International. p. 203. ISBN 81-224-0801-X. http://books.google.com/?id=yZ68h97pnAkC&pg=PA203. 
12. ^ Eduard Prugovec̆ki (1981). "Definition 2.1". Quantum mechanics in Hilbert space (2nd ed.). Academic Press. pp. 18 ff. ISBN 0-12-566060-X. http://books.google.com/?id=GxmQxn2PF3IC&pg=PA18. 
13. ^ Gunawardena, Jeremy. "Matrix algebra for beginners, Part I". Harvard Medical School. http://vcp.med.harvard.edu/papers/matrices-1.pdf. Retrieved 2 May 2012. 
14. ^ Miller, Steven. "The Method of Least Squares". Brown University. http://web.williams.edu/go/math/sjmiller/public_html/BrownClasses/54/handouts/MethodLeastSquares.pdf. Retrieved 3 May 2012. 

Further reading[link]

External links[link]

Wikibooks has a book on the topic of Linear Algebra

• International Linear Algebra Society
• MIT Professor Gilbert Strang's Linear Algebra Course Homepage : MIT Course Website
• MIT Linear Algebra Lectures: free videos from MIT OpenCourseWare
• Linear Algebra Toolkit.
• Linear Algebra on MathWorld.
• Linear Algebra overview and notation summary on PlanetMath.
• Linear Algebra tutorial with online interactive programs.
• Matrix and Linear Algebra Terms on Earliest Known Uses of Some of the Words of Mathematics
• Earliest Uses of Symbols for Matrices and Vectors on Earliest Uses of Various Mathematical Symbols
• Linear Algebra by Elmer G. Wiens. Interactive web pages for vectors, matrices, linear equations, etc.
• Linear Algebra Solved Problems: Interactive forums for discussion of linear algebra problems, from the lowest up to the hardest level (Putnam).
• Linear Algebra for Informatics. José Figueroa-O'Farrill, University of Edinburgh
• Online Notes / Linear Algebra Paul Dawkins, Lamar University
• Elementary Linear Algebra textbook with solutions
• Linear Algebra Wiki
• Linear algebra (math 21b) homework and exercises
• Textbook and solutions manual, Saylor Foundation.

Online books[link]

• Beezer, Rob, A First Course in Linear Algebra
• Connell, Edwin H., Elements of Abstract and Linear Algebra
• Hefferon, Jim, Linear Algebra
• Matthews, Keith, Elementary Linear Algebra
• Sharipov, Ruslan, Course of linear algebra and multidimensional geometry
• Treil, Sergei, Linear Algebra Done Wrong

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In mathematics, a linear map or linear function f(x) is a function which satisfies the following two properties:

• Additivity (also called the superposition property): f(x + y) = f(x) + f(y). This says that f is a group homomorphism with respect to addition.
• Homogeneity of degree 1: f(αx) = αf(x) for all α.

It can be shown that additivity implies the homogeneity in all cases where α is rational; this is done by proving the case where α is a natural number by mathematical induction and then extending the result to arbitrary rational numbers. If f is assumed to be continuous as well then this can be extended to show that homogeneity for α any real number, using the fact that rationals form a dense subset of the reals.

In this definition, x is not necessarily a real number, but can in general be a member of any vector space. A less restrictive definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics.

The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operator, and many constructed from it, such as del and the Laplacian. When a differential equation can be expressed in linear form, it is particularly easy to solve by breaking the equation up into smaller pieces, solving each of those pieces, and adding the solutions up.

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations (also called linear maps), and systems of linear equations.

The word linear comes from the Latin word linearis, which means pertaining to or resembling a line. For a description of linear and nonlinear equations, see linear equation. Nonlinear equations and functions are of interest to physicists and mathematicians because they can be used to represent many natural phenomena, including chaos.

Integral linearity[link]

For a device that converts a quantity to another quantity there are three basic definitions for integral linearity in common use: independent linearity, zero-based linearity, and terminal, or end-point, linearity. In each case, linearity defines how well the device's actual performance across a specified operating range approximates a straight line. Linearity is usually measured in terms of a deviation, or non-linearity, from an ideal straight line and it is typically expressed in terms of percent of full scale, or in ppm (parts per million) of full scale. Typically, the straight line is obtained by performing a least-squares fit of the data. The three definitions vary in the manner in which the straight line is positioned relative to the actual device's performance. Also, all three of these definitions ignore any gain, or offset errors that may be present in the actual device's performance characteristics.

Many times a device's specifications will simply refer to linearity, with no other explanation as to which type of linearity is intended. In cases where a specification is expressed simply as linearity, it is assumed to imply independent linearity.

Independent linearity is probably the most commonly-used linearity definition and is often found in the specifications for DMMs and ADCs, as well as devices like potentiometers. Independent linearity is defined as the maximum deviation of actual performance relative to a straight line, located such that it minimizes the maximum deviation. In that case there are no constraints placed upon the positioning of the straight line and it may be wherever necessary to minimize the deviations between it and the device's actual performance characteristic.

Zero-based linearity forces the lower range value of the straight line to be equal to the actual lower range value of the device's characteristic, but it does allow the line to be rotated to minimize the maximum deviation. In this case, since the positioning of the straight line is constrained by the requirement that the lower range values of the line and the device's characteristic be coincident, the non-linearity based on this definition will generally be larger than for independent linearity.

For terminal linearity, there is no flexibility allowed in the placement of the straight line in order to minimize the deviations. The straight line must be located such that each of its end-points coincides with the device's actual upper and lower range values. This means that the non-linearity measured by this definition will typically be larger than that measured by the independent, or the zero-based linearity definitions. This definition of linearity is often associated with ADCs, DACs and various sensors.

A fourth linearity definition, absolute linearity, is sometimes also encountered. Absolute linearity is a variation of terminal linearity, in that it allows no flexibility in the placement of the straight line, however in this case the gain and offset errors of the actual device are included in the linearity measurement, making this the most difficult measure of a device's performance. For absolute linearity the end points of the straight line are defined by the ideal upper and lower range values for the device, rather than the actual values. The linearity error in this instance is the maximum deviation of the actual device's performance from ideal.

Linear polynomials[link]

In a different usage to the above, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a line.

Over the reals, a linear equation is one of the forms:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = m x + b\

where m is often called the slope or gradient; b the y-intercept, which gives the point of intersection between the graph of the function and the y-axis.

Note that this usage of the term linear is not the same as the above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so if and only if b = 0. Hence, if b ≠ 0, the function is often called an affine function (see in greater generality affine transformation).

Boolean functions[link]

In Boolean algebra, a linear function is a function Failed to parse (Missing texvc executable; please see math/README to configure.): f

for which there exist Failed to parse (Missing texvc executable; please see math/README to configure.): a_0, a_1, \ldots, a_n \in \{0,1\}
such that 

Failed to parse (Missing texvc executable; please see math/README to configure.): f(b_1, \ldots, b_n) = a_0 \oplus (a_1 \land b_1) \oplus \cdots \oplus (a_n \land b_n)

for all Failed to parse (Missing texvc executable; please see math/README to configure.): b_1, \ldots, b_n \in \{0,1\}.

A Boolean function is linear if one of the following holds for the function's truth table:

1. In every row in which the truth value of the function is 'T', there are an odd number of 'T's assigned to the arguments and in every row in which the function is 'F' there is an even number of 'T's assigned to arguments. Specifically, f('F', 'F', ..., 'F') = 'F', and these functions correspond to linear maps over the Boolean vector space.
2. In every row in which the value of the function is 'T', there is an even number of 'T's assigned to the arguments of the function; and in every row in which the truth value of the function is 'F', there are an odd number of 'T's assigned to arguments. In this case, f('F', 'F', ..., 'F') = 'T'.

Another way to express this is that each variable always makes a difference in the truth-value of the operation or it never makes a difference.

Negation, Logical biconditional, exclusive or, tautology, and contradiction are linear functions.

Physics[link]

In physics, linearity is a property of the differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation.

Linearity of a differential equation means that if two functions f and g are solutions of the equation, then any linear combination af+bg is too.

Electronics[link]

In electronics, the linear operating region of a device, for example a transistor, is where a dependent variable (such as the transistor collector current) is directly proportional to an independent variable (such as the base current). This ensures that an analog output is an accurate representation of an input, typically with higher amplitude (amplified). A typical example of linear equipment is a high fidelity audio amplifier, which must amplify a signal without changing its waveform. Others are linear filters, linear regulators, and linear amplifiers in general.

In most scientific and technological, as distinct from mathematical, applications, something may be described as linear if the characteristic is approximately but not exactly a straight line; and linearity may be valid only within a certain operating region—for example, a high-fidelity amplifier may distort even a small signal, but sufficiently little to be acceptable (acceptable but imperfect linearity); and may distort very badly if the input exceeds a certain value, taking it away from the approximately linear part of the transfer function.

Military tactical formations[link]

In military tactical formations, "linear formations" were adapted from phalanx-like formations of pike protected by handgunners towards shallow formations of handgunners protected by progressively fewer pikes. This kind of formation would get thinner until its extreme in the age of Wellington with the 'Thin Red Line'. It would eventually be replaced by skirmish order at the time of the invention of the breech-loading rifle that allowed soldiers to move and fire independently of the large scale formations and fight in small, mobile units.

Art[link]

Linear is one of the five categories proposed by Swiss art historian Heinrich Wölfflin to distinguish "Classic", or Renaissance art, from the Baroque. According to Wölfflin, painters of the fifteenth and early sixteenth centuries (Leonardo da Vinci, Raphael or Albrecht Dürer) are more linear than "painterly" Baroque painters of the seventeenth century (Peter Paul Rubens, Rembrandt, and Velázquez) because they primarily use outline to create shape.^[1] Linearity in art can also be referenced in digital art. For example, hypertext fiction can be an example of nonlinear narrative, but there are also websites designed to go in a specified, organized manner, following a linear path.

Music[link]

In music the linear aspect is succession, either intervals or melody, as opposed to simultaneity or the vertical aspect.

Measurement[link]

In measurement, the term "linear foot" refers to the number of feet in a straight line of material (such as lumber or fabric) generally without regard to the width. It is sometimes incorrectly referred to as "lineal feet"; however, "lineal" is typically reserved for usage when referring to ancestry or heredity.[1] The words "linear"[2] & "lineal" [3] both descend from the same root meaning, the Latin word for line, which is "linea".

See also[link]

• Linear element
• Linear system
• Linear medium
• Linear programming
• Bilinear
• Multilinear
• Linear motor
• Linear A and Linear B scripts.
• Linear interpolation

References[link]

1. ^ Heinrich Wölfflin, Principles of Art History: the Problem of the Development of Style in Later Art, M. D. Hottinger (trans.), Mineola, N.Y.: Dover (1950): pp. 18–72.