 Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps (or linear transformations or linear operators) and can be represented by matrices if a basis is given. Thus matrix theory is often considered as a part of linear algebra. Linear algebra is commonly restricted to the case of finite dimensional vector spaces, while the peculiarities of the infinite dimensional case are traditionally covered in linear functional analysis.
Linear algebra is central to modern mathematics and its applications. An elementary application of linear algebra is to find the solution of a system of linear equations in several unknowns. More advanced applications are ubiquitous in areas as diverse as abstract algebra and functional analysis. Linear algebra has a concrete representation in analytic geometry and is generalized in operator theory and in module theory. It has extensive applications in engineering, physics, natural sciences, computer science, and the social sciences (particularly in economics). Nonlinear mathematical models can often be approximated by linear ones.
Contents
History
The subject first took its modern form in the first half of the twentieth century. At this time, many ideas and methods of previous centuries were generalized as abstract algebra. Matrices and tensors were introduced in the latter part of the 19th century. The use of these objects in quantum mechanics, special relativity, and statistics did much to spread the subject of linear algebra beyond pure mathematics.
The origin of many of these ideas is discussed in the articles on determinants and Gaussian elimination.
Main structures
The main structures of linear algebra are vector spaces and linear maps between them. A vector space is a set whose elements can be added together and multiplied by the scalars, or numbers. In many physical applications, the scalars are real numbers, R. More generally, the scalars may form any field F—thus one can consider vector spaces over the field Q of rational numbers, the field C of complex numbers, or a finite field F_{q}.
In a vector space, the operations of addition and scalar multiplication must behave similarly to the usual addition and multiplication of numbers: addition is commutative and associative, multiplication distributes over addition, and so on. More precisely, the two operations must satisfy a list of axioms chosen to emulate the properties of addition and scalar multiplication of Euclidean vectors in the coordinate nspace R^{n}. One of the axioms stipulates the existence of a zero vector, which behaves analogously to the number zero with respect to addition.
Elements of a general vector space V may be objects of any nature, for example, functions or polynomials, but when viewed as elements of V, they are frequently called vectors.
Given two vector spaces V and W over a field F, a linear transformation (or "linear map") is a map
that is compatible with addition and scalar multiplication:
for any vectors u,v ∈ V and a scalar r ∈ F.
Other fundamental notions in linear algebra include: linear combination, span, linear independence of vectors, a basis of a vector space, and the dimension of a vector space.
Given a vector space V over a field F, an expression of the form
where v_{1}, v_{2}, …, v_{k} are vectors and r_{1}, r_{2}, …, r_{k} are scalars, is called the linear combination of the vectors v_{1}, v_{2}, …, v_{k} with coefficients r_{1}, r_{2}, …, r_{k}. The set of all linear combinations of vectors v_{1}, v_{2}, …, v_{k} is called their span. A linear combination of any system of vectors with all zero coefficients is 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. A linearly independent set of vectors that spans a vector space V is a basis of V. If a vector space admits a finite basis then any two bases have the same number of elements (called the dimension of V) and V is a finitedimensional vector space. This theory can be extended to infinitedimensional spaces.
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 a basis and many different bases exist (although they all consist of the same number of elements equal to the dimension of V). Having a particular basis {v_{1}, v_{2}, …, v_{n}} of V allows one to construct a coordinate system in V: the vector with coordinates (r_{1}, r_{2}, …, r_{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} further assures that these coordinates are determined in a unique way (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. Therefore, by and large, the study of linear transformations, which were defined axiomatically, may be replaced by the study of matrices, which are concrete objects. This is a major technique in linear algebra.
Some main useful theorems
 (AC) Every vector space has a basis.^{[1]}
 (AC) Any two bases of the same vector space have the same cardinality. Equivalently, the dimension of a vector space is welldefined.^{[2]}
 A matrix is invertible, or nonsingular, 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 F^{n} as a vector space over F.
 Corollary: Any two vector spaces over F of the same finite dimension are isomorphic to each other.
Since linear algebra is a successful theory, its methods have been developed in other parts of mathematics. In module theory, one replaces the field of scalars by a ring. 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 tensor product. Functional analysis mixes the methods of linear algebra with those of mathematical analysis.
See also
 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
 ^ The existence of a basis is straightforward for countably generated vector spaces, and for wellordered vector spaces, but in full generality it is logically equivalent to the axiom of choice.
 ^ Dimension theorem for vector spaces
Further reading
 History
 FearnleySander, Desmond, "Hermann Grassmann and the Creation of Linear Algebra" (via JSTOR), American Mathematical Monthly 86 (1979), pp. 809–817.
 Grassmann, Hermann, Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert, O. Wigand, Leipzig, 1844.
 Introductory textbooks
 Axler, Sheldon (February 26, 2004), Linear Algebra Done Right (2nd ed.), Springer, ISBN 9780387982588
 Bretscher, Otto (June 28, 2004), Linear Algebra with Applications (3rd ed.), Prentice Hall, ISBN 9780131453340
 Farin, Gerald; Hansford, Dianne (December 15, 2004), Practical Linear Algebra: A Geometry Toolbox, AK Peters, ISBN 9781568812342
 Friedberg, Stephen H.; Insel, Arnold J.; Spence, Lawrence E. (November 11, 2002), Linear Algebra (4th ed.), Prentice Hall, ISBN 9780130084514
 Hefferon, Jim (2008), Linear Algebra, http://joshua.smcvt.edu/linearalgebra/
 Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
 Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 9780321287137
 Kolman, Bernard; Hill, David R. (May 3, 2007), Elementary Linear Algebra with Applications (9th ed.), Prentice Hall, ISBN 9780132296540
 Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall, ISBN 9780131857858
 Poole, David (2010), Linear Algebra: A Modern Introduction (3rd ed.), Cengage  Brooks/Cole, ISBN 9780538735452
 Ricardo, Henry (2010), A Modern Introduction To Linear Algebra (1st ed.), CRC Press, ISBN 9781439800409
 Strang, Gilbert (July 19, 2005), Linear Algebra and Its Applications (4th ed.), Brooks Cole, ISBN 9780030105678
 Advanced textbooks
 Bhatia, Rajendra (November 15, 1996), Matrix Analysis, Graduate Texts in Mathematics, Springer, ISBN 9780387948461
 Demmel, James W. (August 1, 1997), Applied Numerical Linear Algebra, SIAM, ISBN 9780898713893
 Gantmacher, F.R. (2005, 1959 edition), Applications of the Theory of Matrices, Dover Publications, ISBN 9780486445540
 Gantmacher, Felix R. (1990), Matrix Theory Vol. 1 (2nd ed.), American Mathematical Society, ISBN 9780821813768
 Gantmacher, Felix R. (2000), Matrix Theory Vol. 2 (2nd ed.), American Mathematical Society, ISBN 9780821826645
 Gelfand, I. M. (1989), Lectures on Linear Algebra, Dover Publications, ISBN 9780486660820
 Glazman, I. M.; Ljubic, Ju. I. (2006), FiniteDimensional Linear Analysis, Dover Publications, ISBN 9780486453323
 Golan, Johnathan S. (January 2007), The Linear Algebra a Beginning Graduate Student Ought to Know (2nd ed.), Springer, ISBN 9781402054945
 Golan, Johnathan S. (August 1995), Foundations of Linear Algebra, Kluwer, ISBN 0792336143
 Golub, Gene H.; Van Loan, Charles F. (October 15, 1996), Matrix Computations, Johns Hopkins Studies in Mathematical Sciences (3rd ed.), The Johns Hopkins University Press, ISBN 9780801854149
 Greub, Werner H. (October 16, 1981), Linear Algebra, Graduate Texts in Mathematics (4th ed.), Springer, ISBN 9780801854149
 Hoffman, Kenneth; Kunze, Ray (April 25, 1971), Linear Algebra (2nd ed.), Prentice Hall, ISBN 9780135367971
 Halmos, Paul R. (August 20, 1993), FiniteDimensional Vector Spaces, Undergraduate Texts in Mathematics, Springer, ISBN 9780387900933
 Horn, Roger A.; Johnson, Charles R. (February 23, 1990), Matrix Analysis, Cambridge University Press, ISBN 9780521386326
 Horn, Roger A.; Johnson, Charles R. (June 24, 1994), Topics in Matrix Analysis, Cambridge University Press, ISBN 9780521467131
 Lang, Serge (March 9, 2004), Linear Algebra, Undergraduate Texts in Mathematics (3rd ed.), Springer, ISBN 9780387964126
 Marcus, Marvin; Minc, Henryk (2010), A Survey of Matrix Theory and Matrix Inequalities, Dover Publications, ISBN 9780486671024
 Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 9780898714548, http://www.matrixanalysis.com/DownloadChapters.html
 Mirsky, L. (1990), An Introduction to Linear Algebra, Dover Publications, ISBN 9780486664347
 Roman, Steven (March 22, 2005), Advanced Linear Algebra, Graduate Texts in Mathematics (2nd ed.), Springer, ISBN 9780387247663
 Shilov, Georgi E. (June 1, 1977), Linear algebra, Dover Publications, ISBN 9780486635187
 Shores, Thomas S. (December 6, 2006), Applied Linear Algebra and Matrix Analysis, Undergraduate Texts in Mathematics, Springer, ISBN 9780387331942
 Smith, Larry (May 28, 1998), Linear Algebra, Undergraduate Texts in Mathematics, Springer, ISBN 9780387984551
 Study guides and outlines
 Leduc, Steven A. (May 1, 1996), Linear Algebra (Cliffs Quick Review), Cliffs Notes, ISBN 9780822053316
 Lipschutz, Seymour; Lipson, Marc (December 6, 2000), Schaum's Outline of Linear Algebra (3rd ed.), McGrawHill, ISBN 9780071362009
 Lipschutz, Seymour (January 1, 1989), 3,000 Solved Problems in Linear Algebra, McGrawHill, ISBN 9780070380233
 McMahon, David (October 28, 2005), Linear Algebra Demystified, McGrawHill Professional, ISBN 9780071465793
 Zhang, Fuzhen (April 7, 2009), Linear Algebra: Challenging Problems for Students, The Johns Hopkins University Press, ISBN 9780801891250
External links
 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.
 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 FigueroaO'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
Online books
 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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