Linear algebra

In the three-dimensional Euclidean space, planes represent solutions of linear equations and their intersections represent the common solutions

Linear algebra is the branch of mathematics concerning linear equations such as

a_{1}x_{1}+\cdots +a_{n}x_{n}=b,

linear functions such as

(x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\ldots +a_{n}x_{n},

and their representations through matrices and vector spaces.^[1]^[2]^[3]

Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis may be basically viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and engineering areas, because it allows modeling many natural phenomena, and efficiently computing with such models. For nonlinear systems, which cannot be modeled with linear algebra, linear algebra is often used as a first-order approximation.

History[edit]

Systems of linear equations arose in Europe with the introduction in 1637 by René Descartes of coordinates in geometry. In fact, in this new geometry, now called Cartesian geometry, lines and planes are represented by linear equations, and computing their intersections amounts solving systems of linear equations.

The first systematic methods for solving linear systems used determinants, first considered by Leibniz in 1693. In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's Rule. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, which was initially listed as an advancement in geodesy.^[4]

The study of matrix algebra first emerged in England in the mid-1800s. 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".^[4]

In 1882, Hüseyin Tevfik Pasha wrote the book titled "Linear Algebra".^[5]^[6] 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 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 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.^[4]

Vector spaces[edit]

Until 19th century, linear algebra was introduced through systems of linear equations and matrices. In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic, more general (not limited to the finite-dimensional case), and conceptually simpler, although more abstract.

A vector space over a field $F$ (often the field of the real numbers) is a set $V$ equipped with two binary operations satisfying the following axioms. 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, scalar multiplication, takes any scalar $a$ and any vector $v$ and outputs a new vector $av$ . The axioms that addition and scalar multiplication must satisfy are the following (in the list below, $u, v$ and $w$ are arbitrary elements of $V$ , and $a$ and $b$ are arbitrary scalars in the field $F$ .^[7]

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 $0$ in $V$ , called the zero vector (or simply zero), such that $v + 0 = v$ for all $v$ in $V$ .
Inverse elements of addition	For every $v$ in $V$ , there exists an element $- v$ in $V$ , 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	$1 v = v$ , where $1$ denotes the multiplicative identity of $F$ .

The first four axioms mean that $V$ is an abelian group under addition.

Elements of a vector space may have various nature; for example, they can be sequences, functions, polynomials or matrices. Linear algebra is concerned with properties common to all vector spaces.

Linear maps[edit]

Linear maps are mappings between vector spaces that preserve the vector-space structure. Given two vector spaces $V$ and $W$ over a field $F$ , a linear map (also called, in some contexts, linear transformation, linear mapping or linear operator) is a map

T:V\to W

that is compatible with addition and scalar multiplication, that is

T(u+v)=T(u)+T(v),\quad T(av)=aT(v)

for any vectors $u, v$ in $V$ and scalar $a$ in $F$ .

This implies that for any vectors $u, v$ in $V$ and scalars $a, b$ in $F$ , one has

T(au+bv)=T(au)+T(bv)=aT(u)+bT(v)

When a bijective linear map exists between two vector spaces (that is, every vector from the second space is associated with exactly one in the first), 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, 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 (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map. All these questions can be solved by using Gaussian elimination or some variant of this algorithm.

Subspaces, span, and basis[edit]

The study of subsets of vector spaces that are themselves vector spaces for the induced operations is fundamental, similarly as for many mathematical structures. These subsets are called linear subspaces. More precisely, a linear subspace of a vector space $V$ over a field $F$ is a subset $W$ of $V$ such that $u + v$ and $au$ are in $W$ , for every $u$ , $v$ in $W$ , and every $a$ in $F$ . (These conditions suffices for implying that $W$ is a vector space.)

For example, the image of a linear map, and the inverse image of 0 by a linear map (called kernel or null space) are linear subspaces.

Another important way of forming a subspace is to consider linear combinations of a set $S$ of vectors: the set of all sums

a_{1}v_{1}+a_{2}v_{2}+\cdots +a_{k}v_{k},

where $v 1, v 2, ..., v k$ are in $V$ , and $a 1, a 2, ..., a k$ are in $F$ form a linear subspace called the span of $S$ . The span of $S$ is also the intersection of all linear subspaces containing $S$ . In other words, it is the (smallest for the inclusion relation) linear subspace containing $S$ .

A set of vectors is linearly independent if none is in the span of the others. Equivalently, a set $S$ of vector is linearly independent if the only way to express the zero vector as a linear combination of elements of $S$ is to take zero for every coefficient $a_{i}.$

A set of vectors that spans a vector space is called a spanning set or generating set. If a spanning set $S$ is linearly dependent (that is not linearly independent), then some element $w$ of $S$ is in the span of the other elements of $S$ , and the span would remain the same if one remove $w$ from $S$ . One may continue to remove elements of $S$ until getting a linearly independent spanning set. Such a linearly independent set that spans a vector space $V$ is called a basis of $V$ . The importance of bases lies in the fact that there are together minimal generating sets and maximal independent sets. More precisely, if $S$ is a linearly independent set, and $T$ is a spanning set such that $S\subseteq T,$ then there is a basis $B$ such that $S\subseteq B\subseteq T.$

Any two bases of a vector space $V$ have the same cardinality, which is called the dimension of $V$ ; this is the dimension theorem for vector spaces. Moreover, two vector spaces are isomorphic if and only they have the same dimension.^[8]

If any basis of $V$ (and therefore every basis) has a finite number of elements, $V$ is a finite-dimensional vector space. If $U$ is a subspace of $V$ , then $dim U \leq dim V$ . In the case where $V$ is finite-dimensional, the equality of the dimensions implies $U = V$ .

If U₁ and U₂ are subspaces of V, then

\dim(U_{1}+U_{2})=\dim U_{1}+\dim U_{2}-\dim(U_{1}\cap U_{2}),

where $U_{1}+U_{2}$ denotes the span of $U_{1}\cup U_{2}.$ ^[9]

Matrices[edit]

Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps. Their theory is thus an essential part of linear algebra.

Let $V$ be a finite-dimensional vector space over a field $F$ , and $(v 1, v 2, ..., v m)$ be a basis of $V$ (thus $m$ is the dimension of $V$ ). By definition of a basis, the map

{\begin{aligned}(a_{1},\ldots ,a_{m})&\mapsto a_{1}v_{1}+\cdots a_{m}v_{m}\\F^{m}&\to V\end{aligned}}

is a bijection from $F^{m},$ the set of the sequences of $m$ elements of $F$ , onto $V$ . This is an isomorphism of vector spaces, if $F^{m}$ is equipped of its standard structure of vector space, where vector addition and scalar multiplication are done component by component.

This isomorphism allows representing a vector by its inverse image under this isomorphism, that is by the coordinates vector $(a_{1},\ldots ,a_{m})$ or by the column matrix

{\begin{bmatrix}a_{1}\\\vdots \\a_{m}\end{bmatrix}}.

If $W$ is another finite dimensional vector space (possibly the same), with a basis $(w_{1},\ldots ,w_{n}),$ a linear map $f$ from $W$ to $V$ is well defined by its values on the basis elements, that is $(f(w_{1}),\ldots ,f(w_{n})).$ Thus, $f$ is well represented by the list of the corresponding column matrices. That is, if

f(w_{j})=a_{1,j}v_{1}+\cdots +a_{m,j}v_{m},

for $j = 1, ..., n$ , then $f$ is represented by the matrix

{\begin{bmatrix}a_{1,1}&\ldots &a_{1,n}\\\vdots &\ldots &\vdots \\a_{m,1}&\ldots &a_{m,n}\end{bmatrix}},

with $m$ rows and $n$ columns.

Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition 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. Equivalently, two matrices are similar if one can transform one in the other by elementary row and column operations. For a matrix representing a linear map from $W$ to $V$ , the row operations correspond to change of bases in $V$ and the column operations correspond to change of bases in $W$ . Every matrix is similar to an identity matrix possibly bordered by zero rows and zero columns. In terms of vector space, this means that, for any linear map from $W$ to $V$ , there are bases such a part of the basis of $W$ is mapped bijectively on a part of the basis of $V$ , and that the remaining basis elements of $W$ , if any, are mapped to zero (this is a way of expressing the fundamental theorem of linear algebra). Gaussian elimination is the basic algorithm for finding these elementary operations, and proving this theorem.

Linear systems[edit]

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

{\begin{alignedat}{7}2x&&\;+\;&&y&&\;-\;&&z&&\;=\;&&8\\-3x&&\;-\;&&y&&\;+\;&&2z&&\;=\;&&-11\\-2x&&\;+\;&&y&&\;+\;&&2z&&\;=\;&&-3\end{alignedat}}\qquad {\text{(S)}}

be a linear system.

To such a system, one may associate its matrix

M\left[{\begin{array}{rrr}2&1&-1\\-3&-1&2\\-2&1&2\end{array}}\right]{\text{.}}

and its right member vector

v={\begin{bmatrix}8\\-11\\-3\end{bmatrix}}.

Let $T$ be the linear transformation associated to the matrix $M$ . A solution of the system $(S)$ is a vector

X={\begin{bmatrix}x\\y\\z\end{bmatrix}}

such that

T(X)=v,

that is an element of the preimage of $v$ by $T$ .

Let $(S')$ be the associated homogeneous system, where the right-hand sides of the equations are put to zero. The solutions of $(S')$ are exactly the elements of the kernel of $T$ or, equivalently, $M$ .

The Gaussian-elimination consists of performing elementary row operations on the augmented matrix

M\left[{\begin{array}{rrr|r}2&3&-1&8\\-3&-1&2&-11\\-2&1&2&-3\end{array}}\right]

for putting it in reduced row echelon form. These row operations do not change the set of solutions of the system of equations. In the example, the reduced echelon form is

M\left[{\begin{array}{rrr|r}1&0&0&2\\0&1&0&3\\0&0&1&-1\end{array}}\right],

showing that the system $(S)$ has the unique solution

{\begin{aligned}x&=2\\y&=3\\z&=-1.\end{aligned}}

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, kernels, matrix inverses.

Endomorphisms and square matrices[edit]

A linear endomorphism is a linear map that maps a vector space $V$ to itself. If $V$ has a basis of $n$ elements, such an endomorphism is represented by a square matrix of size $n$ .

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 transformations, coordinate changes, quadratic forms, and many other part of mathematics.

Determinant[edit]

The determinant of a square matrix is a polynomial function of the entries of the matrix, such that the matrix is invertible if and only if the determinant is not zero. This results from the fact that the determinant of a product of matrices is the product of the determinants, and thus that a matrix is invertible if and only if its determinant is invertible.

Cramer's rule is a closed-form expression, in terms of determinants, of the solution of a system of $n$ linear equations in $n$ unknowns. Cramer's rule is useful for reasoning about the solution, but, except for $n = 2$ or $3$ , 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[edit]

If $f$ is a linear endomorphism of a vector space $V$ over a field $F$ , an eigenvector of $f$ is a nonzero vector $v$ of $V$ such that $f (v) = av$ for some scalar $a$ in $F$ . This scalar $a$ is an eigenvalue of $f$ .

If the dimension of $V$ is finite, and a basis has been chosen, $f$ and $v$ may be represented, respectively, by a square matrix $M$ and a column matrix and $z$ ; the equation defining eigenvectors and eigenvalues becomes

Mz=az.

Using the identity matrix $I$ , whose all entries are zero, except those of the main diagonal, which are equal to one, this may be rewritten

(M-aI)z=0.

As $z$ is supposed to be nonzero, this means that $M - aI$ is a singular matrix, and thus that its determinant $\det(M-aI)$ equals zero. The eigenvalues are thus the roots of the polynomial

\det(xI-M).

If $V$ is of dimension $n$ , this is a monic polynomial of degree $n$ , called the characteristic polynomial of the matrix (or of the endomorphism), and there are, at most, $n$ eigenvalues.

If a basis exists that consists only of eigenvectors, the matrix of $f$ on this basis has a very simple structure: it is a diagonal matrix such that the entries on the main diagonal are eigenvalues, and the other entries are zero. In this case, the endomorphism and the matrix are said diagonalizable. 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, then the matrix is diagonalizable.

A symmetric matrix is always diagonalizable. There are non-diagonizable matrices, the simplest being

{\begin{bmatrix}0&1\\0&0\end{bmatrix}}

(it cannot be diagonalizable since its square is the zero matrix, 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 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[edit]

A linear form is a linear map from a vector space $V$ over a field $F$ to the field of scalars $F$ , viewed as a vector space over itself. Equipped by pointwise addition and multiplication by a scalar, the linear forms form a vector space, called the dual space of $V$ , and usually denoted $V^{*}.$

If $v_{1},\ldots ,v_{n}$ is a basis of $V$ (this implies that $V$ is finite-dimensional), then one can define, for $i = 1, ..., n$ , a linear map $v_{i}^{*}$ such that $v_{i}^{*}(e_{i})=1$ and $v_{i}^{*}(e_{j})=0$ if $j \neq i$ . These linear maps form a basis of $V^{*},$ called the dual basis of $v_{1},\ldots ,v_{n}.$ (If $V$ is not finite-dimensional, the $v_{i}^{*}$ may be defined similarly; they are linearly independent, but do not form a basis.)

For $v$ in $V$ , the map

f\to f(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 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

\langle f,x\rangle

for denoting $f (x)$ .

Dual map[edit]

Let

f:V\to W

be a linear map. For every linear form $h$ on $W$ , the composite function $f \circ h$ is a linear form on $V$ . This defines a linear map

f^{*}:W^{*}\to V^{*}

between the dual spaces, which is called the dual or the transpose of $f$ .

If $V$ and $W$ are finite dimensional, and $M$ is the matrix of $f$ in terms of some ordered bases, then the matrix of $f^{*}$ over the dual bases is the transpose $M^{\mathsf {T}}$ of $M$ , 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 by

\langle h^{\mathsf {T}},Mv\rangle =\langle h^{\mathsf {T}}M,v\rangle .

For highlighting this symmetry, the two members of this equality are sometimes written

\langle h^{\mathsf {T}}\mid M\mid v\rangle .

Inner-product spaces[edit]

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

\langle \cdot ,\cdot \rangle :V\times V\rightarrow F

that satisfies the following three axioms for all vectors u, v, w in V and all scalars a in F:^[10]^[11]

Conjugate symmetry:

\langle u,v\rangle ={\overline {\langle v,u\rangle }}.

Note that in R, it is symmetric.

Linearity in the first argument:

\langle au,v\rangle =a\langle u,v\rangle .

\langle u+v,w\rangle =\langle u,w\rangle +\langle v,w\rangle .

Positive-definiteness:

\langle v,v\rangle \geq 0

with equality only for v = 0.

We can define the length of a vector v in V by

\|v\|^{2}=\langle v,v\rangle ,

and we can prove the Cauchy–Schwarz inequality:

|\langle u,v\rangle |\leq \|u\|\cdot \|v\|.

In particular, the quantity

{\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 $\langle u,v\rangle =0$ . 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 easy to deal with, since if v = a₁ v₁ + ... + a_n v_n, then $a_{i}=\langle v,v_{i}\rangle$ .

The inner product facilitates the construction of many useful concepts. For instance, given a transform T, we can define its Hermitian conjugate T* as the linear transform satisfying

\langle Tu,v\rangle =\langle u,T^{*}v\rangle .

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.

Relationship with geometry[edit]

There is a strong relationship between linear algebra and geometry, which started with the introduction by René Descartes, in 1637, of Cartesian coordinates. In this new (at that time) geometry, now called Cartesian geometry, 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 and planes are represented by linear equations. Thus, computing intersections of lines and planes amounts solving systems of linear equations. This was one of the main motivations for developing linear algebra.

Most geometric transformation, such as translations, rotations, reflections, rigid motions, and isometries, 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 homographies and Möbius transformations, when considered as transformations of a projective space.

Until the end of 19th century, geometric spaces were defined by axioms 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 bee shown that the two approaches are essentially equivalent.^[12] 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.

Applications[edit]

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.

Least-squares best-fit line[edit]

The least squares method is used to determine the best-fit line for a set of data.^[13] This line will minimize the sum of the squares of the residuals.

Fourier series expansion[edit]

Fourier series are a representation of a function f: [−π, π] → R as a trigonometric series:

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; all Lipschitz-continuous functions have a converging Fourier series expansion, and some 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

\langle f,g\rangle ={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)g(x)\,dx.

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 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 a_k, we take the inner product with h_k:

\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, $\langle f,h_{k}\rangle =a_{k}$ ; that is,

a_{k}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)\cos(kx)\,dx.

Quantum mechanics[edit]

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 φ: R³ → C such that $\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }|\phi |^{2}dxdydz$ is finite), and it evolves according to the Schrödinger equation. Energy is represented as the operator $H=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(x,y,z)$ , where V is the potential energy. H is also known as the Hamiltonian operator. The eigenvalues of H represent 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 φ 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[edit]

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 every linearly independent subset of a module can be extended to form a basis, and not every subset of a module that spans 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: Vⁿ → 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 V × V → V, 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 L^p 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.

There are several related topics in the field of computer programming that utilize much of the techniques and theorems linear algebra encompasses and refers to.

Notes[edit]

^ Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC, ISBN 978-1420095388
^ Strang, Gilbert (July 19, 2005), Linear Algebra and Its Applications (4th ed.), Brooks Cole, ISBN 978-0-03-010567-8
^ Weisstein, Eric. "Linear Algebra". From MathWorld--A Wolfram Web Resource. Wolfram. Retrieved 16 April 2012.
^ ^a ^b ^c ^d Vitulli, Marie. "A Brief History of Linear Algebra and Matrix Theory". Department of Mathematics. University of Oregon. Archived from the original on 2012-09-10. Retrieved 2014-07-08.
^ http://www.journals.istanbul.edu.tr/tr/index.php/oba/article/download/9103/8452
^ Hussein Tevfik (21 April 1882). "Linear Algebra". A.H. Boyajian. Retrieved 21 April 2018 – via Internet Archive.
^ Roman 2005, ch. 1, p. 27
^ Axler (2004), p. 55
^ Axler (2204), p. 33
^ 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.
^ Eduard Prugovec̆ki (1981). "Definition 2.1". Quantum mechanics in Hilbert space (2nd ed.). Academic Press. pp. 18 ff. ISBN 0-12-566060-X.
^ Emil Artin (1957) Geometric Algebra Interscience Publishers
^ Miller, Steven. "The Method of Least Squares" (PDF). Brown University. Retrieved 1 May 2013.

^ This axiom is not asserting the associativity of an operation, since there are two operations in question, scalar multiplication: bv; and field multiplication: ab.

External links[edit]

Online Resources[edit]

MIT Linear Algebra Video Lectures, a serie of 34 recorded lectures by professor Gilbert Strang (Spring 2010)
International Linear Algebra Society
Hazewinkel, Michiel, ed. (2001) [1994], "Linear algebra", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Linear Algebra on MathWorld.
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
Essence of linear algebra, a video presentation of the basics of linear algebra, with emphasis on the relationship between the geometric, the matrix and the abstract points of view

Online books[edit]

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

[1] Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC, ISBN 978-1420095388

[2] Strang, Gilbert (July 19, 2005), Linear Algebra and Its Applications (4th ed.), Brooks Cole, ISBN 978-0-03-010567-8

[3] Weisstein, Eric. "Linear Algebra". From MathWorld--A Wolfram Web Resource. Wolfram. Retrieved 16 April 2012.

[Vitulli,_Marie-4] Vitulli, Marie. "A Brief History of Linear Algebra and Matrix Theory". Department of Mathematics. University of Oregon. Archived from the original on 2012-09-10. Retrieved 2014-07-08.

[5] ttp://www.journals.istanbul.edu.tr/tr/index.php/oba/article/download/9103/8452

[6] Hussein Tevfik (21 April 1882). "Linear Algebra". A.H. Boyajian. Retrieved 21 April 2018 – via Internet Archive.

[7] Roman 2005, ch. 1, p. 27

[9] Axler (2004), p. 55

[10] Axler (2204), p. 33

[Jain-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.

[Prugovec̆ki-12] Eduard Prugovec̆ki (1981). "Definition 2.1". Quantum mechanics in Hilbert space (2nd ed.). Academic Press. pp. 18 ff. ISBN 0-12-566060-X.

[13] Emil Artin (1957) Geometric Algebra Interscience Publishers

[Least_Squares-14] Miller, Steven. "The Method of Least Squares" (PDF). Brown University. Retrieved 1 May 2013.

[8] This axiom is not asserting the associativity of an operation, since there are two operations in question, scalar multiplication: bv; and field multiplication: ab.

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v t e Linear algebra
Basic concepts	Scalar Vector Vector space Scalar multiplication Vector projection Linear span Linear map Linear projection Linear independence Linear combination Basis Column space Row space Orthogonality Kernel Eigenvalues and eigenvectors Outer product Inner product space Dot product Transpose Gram–Schmidt process Linear equations
Vector algebra	Cross product Triple product Seven-dimensional cross product
Multilinear algebra	Geometric algebra Exterior algebra Bivector Multivector
Matrices	Block Decomposition Invertible Minor Multiplication Rank Transformation Cramer's rule Gaussian elimination
Algebraic constructions	Dual Direct sum Function space Quotient Subspace Tensor product
Numerical	Floating point Matrix Laboratory Numerical stability Basic Linear Algebra Subprograms (BLAS) Sparse matrix Comparison of linear algebra libraries Comparison of numerical analysis software
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Linear algebra

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History[edit]

Vector spaces[edit]

Linear maps[edit]

Subspaces, span, and basis[edit]

Matrices[edit]

Linear systems[edit]

Endomorphisms and square matrices[edit]

Determinant[edit]

Eigenvalues and eigenvectors[edit]

Duality[edit]

Dual map[edit]

Inner-product spaces[edit]

Relationship with geometry[edit]

Applications[edit]

Least-squares best-fit line[edit]

Fourier series expansion[edit]

Quantum mechanics[edit]

Generalizations and related topics[edit]

See also[edit]

Notes[edit]

Further reading[edit]

History[edit]

Introductory textbooks[edit]

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