Linear algebra1 Linear algebra 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 The three-dimensional Euclidean space R3 is a vector space, and lines and planes passing through the origin are vector subspaces in spaces. Combined with calculus linear algebra R3. 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 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. . When a bijective linear mapping exists between two vector spaces (that is. or matrices. for example. called the zero vector. In view of the first example. every vector from the first space is associated with one in the second). Elements of V are called vectors and elements of F are called scalars. called the additive inverse of v. linear mapping or linear operator) is a map that is compatible with addition and scalar multiplication: for any vectors u. Linear transformations Similarly as in the theory of other algebraic structures. linear algebra studies mappings between vector spaces that preserve the vector-space structure.[6] In the list below. Elements of a general vector space V may be objects of any nature. there exists an element −v ∈ V. and a and b scalars in F. vector addition. The first operation. functions. two isomorphic vector spaces are "essentially the same" from the linear algebra point of view. If a mapping is not an isomorphism.Linear algebra 2 Scope of study Vector spaces The main structures of linear algebra are vector spaces. The operations of addition and multiplication in a vector space satisfies following axioms. Because an isomorphism preserves linear structure. vectors. Given two vector spaces V and W over a field F. For every v ∈ V.v ∈ V and a scalar a ∈ F. A vector space over a field F is a set V together with two binary operations. the multiplication is called scalar multiplication of v by a. called the kernel of the mapping. where 1 denotes the multiplicative identity in F. where the multiplication is done by rescaling the vector v by a scalar a. takes any two vectors v and w and outputs a third vector v + w. we say that the two spaces are isomorphic. and this question can be answered by checking if the determinant is nonzero. such that v + 0 = v for all v ∈ V. One essential question in linear algebra is whether a mapping is an isomorphism or not. v and w be arbitrary vectors in V. polynomials. such that v + (−v) = 0 a(u + v) = au + av Distributivity of scalar multiplication with respect to vector addition Distributivity of scalar multiplication with respect to field addition Compatibility of scalar multiplication with field multiplication Identity element of scalar multiplication (a + b)v = av + bv a(bv) = (ab)v [7] 1v = v. The second operation takes any scalar a and any vector v and outputs a new vector vector av. a linear transformation (also called linear map. linear algebra is interested in finding its range (or image) and the set of elements that get mapped to zero. Linear algebra is concerned with properties common to all vector spaces. let u. Axiom Associativity of addition Commutativity of addition Identity element of addition Inverse elements of addition Signification u + (v + w) = (u + v) + w u+v=v+u There exists an element 0 ∈ V. v2. While Rn has a standard basis {e1. …. This major technique distinguishes linear algebra from theories of other algebraic structures. and are thus often called the range space and the nullspace. span. which are concrete objects. vn assures that these coordinates are unique (i. which were defined axiomatically. a set of linearly dependent vectors is redundant in the sense that a linearly independent subset will span the same subspace. vk: where a1. …. V is called a finite-dimensional vector space. these are important examples of subspaces. if V and W are an n-dimensional and m-dimensional vector space over F. v2. and indeed. v2. but no explicit basis has been constructed. and any linearly independent set of vectors in V can be extended to a basis. …. The set of all linear combinations of vectors v1. Furthermore. If U1 and U2 are subspaces of V. Any set of vectors that spans V contains a basis.[10] 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. Vectors as n-tuples: matrix theory A particular basis {v1. If this is the only way to express zero vector as a linear combination of v1.[11] giving an easy way of characterizing isomorphism. e2. …. which is called the dimension of V. then . there is only one linear combination of the basis vectors that is equal to v). a2. called the matrix of T with respect to these bases. vn span V guarantees that each vector v can be assigned coordinates. linear algebra is interested in subsets of vector spaces that are vector spaces themselves. a2. Therefore. V may be identified with the coordinate n-space Fn. an) is the linear combination The condition that v1. …. v2. if any vector is a linear combination of other vectors (and so the set is not linearly independent). may not even be constructable. 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). Given a set of vectors that span a space. then dim U ≤ dim V. …. Another important way of forming a subspace is to take a linear combination of a set of vectors v1. every vector space has a basis. vn} of V allows one to construct a coordinate system in V: the vector with coordinates (a1. which forms a subspace. For instance. Thus. and basis Again in analogue with theories of other algebraic objects. then any linear transformation T: V → W may be encoded by an m × n matrix A with entries in the field F. v2. vk is called their span. by the study of matrices.[9] nevertheless. The dimension of a vector space is well-defined by the dimension theorem for vector spaces.[8] It turns out that if we accept the axiom of choice. In this way. …. vk then these vectors are linearly independent. addition and scalar multiplication of vectors in V correspond to addition and scalar multiplication of their coordinate vectors in Fn. …. then the span would remain the same if we remove from the set. …. once a basis of a vector space V over F has been chosen. which we call a basis of V. there exists a basis for the real numbers considered as a vector space over the rationals. If V is finite-dimensional and U is a subspace of V. these subsets are called linear subspaces. ak are scalars. There is an important distinction between the coordinate n-space Rn and a general finite-dimensional vector space V. which usually cannot be parametrized so concretely. Under this identification. Two matrices that encode the same linear transformation in different bases are called similar. whereas the linear independence of v1.Linear algebra 3 Subspaces. Any two bases of a vector space V have the same cardinality. we are mostly interested in a linearly independent set of vectors that spans a vector space V.e. Matrix theory replaces the study of linear transformations. For instance. A linear combination of any system of vectors with all zero coefficients is the zero vector of V. . this basis may be unnatural. If a basis of V has finite number of elements. and a basis of V and a basis of W have been fixed. the range and kernel of a linear mapping are both subspaces. v2. en}. Thus. We are especially interested in those non-zero vectors such that . action of the transformation on any vector: if (not necessarily taking a vector space consisting of eigenvectors.Linear algebra One major application of the matrix theory is calculation of determinants. Such a transformation is called a diagonalizable matrix since in the eigenbasis. we can easily compute the are linearly independent eigenvectors of a mapping of distinct) eigenvalues . then. 4 Eigenvalues and eigenvectors In general. Determinants could also be used to solve systems of linear equations (see Cramer's rule). If such a basis exists. an inner product is a map :[12][13] that satisfies the following three axioms for all vectors • Conjugate symmetry: and all scalars Note that in R. the value of the determinant does not depend on the specific basis. 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 Inner-product spaces Besides these basic concepts. and so the eigenvalues are not guaranteed to exist if the field is R. It would be particularly nice if given a transformation into itself we can find a basis for n-dimensional spaces with . Gaussian elimination is a faster method. Formally. a central concept in linear algebra. the action of a linear transformation is hard to understand. While determinants could be defined in a basis-free manner. . Determinants have other applications. those vectors that are relatively fixed by that transformation are given special attention. computations involving matrices are much simpler if we can bring the matrix to a diagonal form. It turns out that a mapping is invertible if and only if the determinant is nonzero. To make this more concrete. the vectors are linearly dependent). but diagonalizable matrices form a dense subset of all matrices. and it gives the vector space a geometric structure by allowing for the definition of length and angles. they are usually introduced via a specific representation of the mapping. the transformation is represented by a diagonal matrix. The inner product is an example of a bilinear form. Because operations like matrix multiplication. and the corresponding scalars are called eigenvalues. matrix inversion. linear algebra also studies vector spaces with additional structure. • Linearity in the first argument: . These vectors are called eigenvectors. we note that where is the identity matrix. and so to get a better handle over linear transformations. To find an eigenvector or an eigenvalue. such as an inner product. then the nullspace is nontrivial. Not all matrices are diagonalizable (even over an algebraically closed field). If the determinant is zero. The determinant is a polynomial. where is a scalar in the base field of the vector space. let be any linear transformation. but in real applications. and if the determinant of that matrix is zero. and determinant calculation are simple on diagonal matrices. it is symmetric. For there to be nontrivial solutions to that equation. 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. if and only if the linear map represented by the matrix is an isomorphism. . and we can prove the Cauchy–Schwartz 5 and so we can call this quantity the cosine of the angle between the two vectors. A linear map is an isomorphism if and only if the determinant is nonzero. since if . This will put the system into triangular form. each by adding to . an orthonormal basis could be found by the Gram–Schmidt procedure. we call T normal. we can define its Hermitian conjugate as the linear transform satisfying If T satisfies . x is eliminated from Formally: by adding to . Applications Because of the ubiquity of vector spaces. and social science. then . Corollary: Any two vector spaces over F of the same finite dimension are isomorphic to each other. Then. It turns out that normal matrices are precisely the matrices that have an orthonormal system of eigenvectors that span V. computer science. given a transform . Two vectors are orthogonal if . For instance. Any vector space over a field F of dimension n is isomorphic to Fn as a vector space over F. or non-singular. the quantity by . and then eliminate y . The inner product facilitates the construction of many useful concepts. Some main useful theorems • • • • A matrix is invertible. using back-substitution. In the example. of the following system of linear equations: The Gaussian-elimination algorithm is as follows: eliminate x from all equations below from all equations below unknown can be solved for. x is then eliminated from . if any. linear algebra is used in many fields of mathematics. Below are just some examples of applications of linear algebra.Linear algebra • Positive-definiteness: with equality only for We can define the length of a vector inequality: In particular. Solution of linear systems 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). Given any finite-dimensional vector space. An orthonormal basis is a basis where all basis vectors have length 1 and are orthogonal to each other. Orthonormal bases are particularly nice to deal with. natural sciences. we find the nullspace of A. The solution set of this equation is given by . Then. write any system of linear equations as a matrix equation: The solution of this system is characterized as follows: first. in general. since two different discontinuous functions might have the same Fourier series). we will not be concerned with convergence issues. the equation has a unique solution if and only if . z and y can be substituted into . and so the first part of the algorithm is complete. If the number of variables equal the . that is. It can thus be seen that Then. which can then be solved to obtain Next. The last part. we compute the solutions of . then we can characterize when the system has a unique solution: since N is trivial if and only if Least-squares best fit line The least squares method is used to determine the best fit line for a set of data. The space of all functions that can be represented by a Fourier series form a vector space (technically speaking.[15] This line will minimize the sum of the squares of the residuals. which can be solved to obtain The system is solved.Linear algebra The result is: 6 Now y is eliminated from by adding to : The result is: This result is a system of linear equations in triangular form. Fourier series expansion Fourier series are a representation of a function as a trigonometric series: This series expansion is extremely useful in solving partial differential equations. [14] number of equations. back-substitution. Moreover. we call functions that have the same Fourier series expansion the "same" function. consists of solving for the knowns in reverse order. can be substituted into . this space is also an inner product space with the inner . We can. and nice enough discontinuous functions have a Fourier series that converges to the function value at most points. it is nice to note that all continuous functions have a converging Fourier series expansion. we find a particular solution of this equation using Gaussian elimination. In this article. that is. not all modules have a basis (those that do are called free modules). mappings that are linear in each of a number of different variables. The concept of eigenvalues and eigenvectors is especially important. Given a particle in some state . The component of H in each eigenstate determines the probability of measuring the corresponding eigenvalue. then the vector space is called an algebra. Representation theory studies the actions of algebraic objects on vector spaces by representing these objects as matrices. or the algebra of polynomials). and it evolves according to the Schrödinger equation. to find the coefficient . its methods have been developed and generalized in other parts of mathematics. and the measurement forces the particle to assume that eigenstate (wave function collapse). This line of inquiry naturally leads to the idea of the dual space. More concretely. in addition to vector addition and scalar multiplication. The eigenvalues of H represents the possible energies that can be observed. Quantum mechanics Quantum mechanics is highly inspired by notions in linear algebra. 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. the rank of a free module is not necessarily unique. that is. . and not all subsets of a module that span the space contains a basis. For instance. not all linearly independent subsets of a module can be extended to form a basis. where V is the potential energy. and it does so by finding subspaces invariant under all transformations of the algebra. energy. and dimension (which is called rank in module theory) still make sense. Energy is represented as the operator . we take the inner product with : and by orthonormality. Generalizations and related topics Since linear algebra is a successful theory. . If. In module theory. one replaces the field of scalars by a ring. Functional analysis mixes the methods of linear algebra with those of mathematical analysis and studies various function spaces. one considers multivariable linear transformations. many theorems from linear algebra become false in module theory. the wave function of a particle describes its physical state and lies in the vector space L2 (the functions such that is finite). and angular momentum) are represented by linear operators on the underlying vector space. for instance. The concepts of linear independence. the vector space consisting of linear maps where F is the field of scalars. Nevertheless. we can expand into a linear combination of eigenstates of H. In quantum mechanics. and observables (such as momentum. basis. H is also known as the Hamiltonian operator. It is interested in all the ways that this is possible. For instance. such as Lp spaces. span. there is a bilinear vector product. associative algebras are algebras with an associate vector product (like the algebra of square matrices. the physical state of a particle is represented by a vector. In multilinear algebra.Linear algebra product 7 The functions for and for are an orthonormal basis for the space of Fourier-expandable functions. Multilinear maps can be described via tensor products of elements of . David (2010). Addison Wesley. Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik. com/ ?id=GxmQxn2PF3IC& pg=PA18). "Matrix algebra for beginners. Steven. [8] Axler (2004). Linear Algebra (http://joshua. uoregon. ch. Desmond. html). Retrieved 2 May 2012. since there are two operations in question. Part I" (http:/ / vcp. "The Method of Least Squares" (http:/ / web.). Howard (2005). David R. Gerald.smcvt. (August 22. Bernard. 1844. pp. Functional analysis (2nd ed. 2004). com/ ?id=yZ68h97pnAkC& pg=PA203).pdf)). Pearson Prentice Hall. . Lawrence E.edu/linearalgebra/) • Anton. 203. Hill. harvard. Khalil Ahmad (1995). ISBN 978-0130084514 • Hefferon. die Lehre vom Magnetismus und die Krystallonomie erläutert. pp. Stephen H. ISBN 978-0132296540 • Leon. New Age International. 2005). wie auch auf die Statik.). Brown University. Marie. Marie [6] Roman 2005. Harvard Medical School. Academic Press. Practical Linear Algebra: A Geometry Toolbox. p. Linear Algebra (4th ed. From MathWorld--A Wolfram Web Resource. edu/ ~vitulli/ 441. Wigand. ISBN 978-0538735452 . Mechanik. . Retrieved 16 April 2012. AK Peters. Wiley International • Lay.. wolfram. Dianne (December 15. Elementary Linear Algebra (Applications Version) (9th ed. Hansford. American Mathematical Monthly 86 (1979). Wolfram. Marie [4] Vitulli. Eric. google. 2002). p. p. html). pdf).. [3] Vitulli. (May 3.1 Definitions and basic properties of inner product spaces and Hilbert spaces" (http:/ / books. Prentice Hall. ISBN 012566060X. 2007).). edu/ go/ math/ sjmiller/ public_html/ BrownClasses/ 54/ handouts/ MethodLeastSquares. 55 [12] P. sp04/ LinAlgHistory. Linear Algebra and Its Applications (3rd ed. "Linear Algebra" (http:/ / mathworld.1" (http:/ / books. [10] Axler (2204). pp. com/ LinearAlgebra. Linear Algebra with Applications (3rd ed. Insel. Linear Algebra With Applications (7th ed. . ISBN 812240801X. and for well-ordered vector spaces.).). Spence. 18 ff.). ISBN 978-0131857858 • Poole. p. "Hermann Grassmann and the Creation of Linear Algebra" ( (http://mathdl. [15] Miller. O. Quantum mechanics in Hilbert space (2nd ed. 2004). [14] Gunawardena. 809–817. ISBN 978-0321287137 • Kolman. "5.). Cengage – Brooks/Cole.maa. K. ISBN 978-1568812342 • Friedberg. Linear Algebra: A Modern Introduction (3rd ed. Jim (2008). David C. Jain. University of Oregon. . scalar multiplication: bv. 1.. but in full generality it is logically equivalent to the axiom of choice.Linear algebra 8 Notes [1] Weisstein. ISBN 978-0131453340 • Farin. Leipzig. Arnold J. Prentice Hall. med. google. (2006). Retrieved 01/24/2012. "A Brief History of Linear Algebra and Matrix Theory" (http:/ / darkwing. • Grassmann.). (November 11. Elementary Linear Algebra with Applications (9th ed. Introductory textbooks • Bretscher. 28–29 [9] The existence of a basis is straightforward for countably generated vector spaces. Jeremy. Retrieved 3 May 2012. Hermann. Further reading History • Fearnley-Sander. Otto (June 28. . Department of Mathematics.). and field multiplication: ab. . Steven J. "Definition 2. org/images/upload_library/22/Ford/DesmondFearnleySander. Marie [5] Vitulli. 33 [11] Axler (2004). [13] Eduard Prugovec̆ki (1981). [2] Vitulli. Prentice Hall. pdf). williams. 27 [7] This axiom is not asserting the associativity of an operation. edu/ papers/ matrices-1. 2005). A Survey of Matrix Theory and Matrix Inequalities. Ray (April 25. Springer. ISBN 978-0486635187 • Shores. Serge (March 9. Cambridge University Press.html).). Undergraduate Texts in Mathematics (3rd ed. Charles F. I. 1 (2nd ed.).R. ISBN 978-0135367971 • Halmos. Matrix Computations. Dover Publications. Linear algebra. ISBN 0792336143 • Golub. (August 1995). American Mathematical Society. An Introduction to Linear Algebra. Springer. Undergraduate Texts in Mathematics. ISBN 978-1402054945 • Golan. Society for Industrial and Applied Mathematics (SIAM).). Charles R. Thomas S. Linear Algebra. ISBN 978-0387900933 • Horn. (2006). Springer. (June 1. 1990). Roger A. A Modern Introduction To Linear Algebra (1st ed. Graduate Texts in Mathematics (2nd ed. ISBN 978-0486453323 • Golan.). (February 23. 2006). com/DownloadChapters. Graduate Texts in Mathematics (4th ed. ISBN 978-0387331942 • Smith. 1996). Dover Publications. (2000). (January 2007). 1971). Matrix Theory Vol. 2005). Johnathan S. ISBN 978-0387984551 9 . 1959 edition). Rajendra (November 15. ISBN 978-0387948461 • Demmel. Matrix Analysis and Applied Linear Algebra (http://www.Linear algebra • Ricardo. Henryk (2010). 1977). Sheldon (February 26. Finite-Dimensional Linear Analysis.matrixanalysis. I. Larry (May 28. (August 20. ISBN 978-0486664347 • Roman. Graduate Texts in Mathematics. 2004). Henry (2010). Applied Linear Algebra and Matrix Analysis. Carl D. American Mathematical Society. ISBN 978-0486445540 • Gantmacher. 2001). 1981).. Linear Algebra (2nd ed. Johnathan S. Advanced Linear Algebra. F. Johnson. 1998). ISBN 978-0486660820 • Glazman. Linear Algebra. ISBN 978-0801854149 • Hoffman. Linear Algebra. Kunze. Kluwer. Matrix Analysis. Springer. Steven (March 22. Undergraduate Texts in Mathematics. Foundations of Linear Algebra. Prentice Hall. ISBN 978-1-4398-0040-9 • Strang. Johns Hopkins Studies in Mathematical Sciences (3rd ed. ISBN 978-0521386326 • Horn. ISBN 978-0387982588 • Bhatia. The Johns Hopkins University Press. Minc. CRC Press. 1997). ISBN 978-0801854149 • Greub. Linear Algebra Done Right (2nd ed. ISBN 978-0030105678 Advanced textbooks • Axler. Ju. 1993). Gene H. ISBN 978-0387247663 • Shilov. (October 15. 2 (2nd ed.). Roger A. (August 1. Werner H. 1996). (1989).). Gilbert (July 19. L. Lectures on Linear Algebra.. (1990). Felix R. Springer. Linear Algebra and Its Applications (4th ed. Matrix Analysis. M. Brooks Cole.. ISBN 978-0821813768 • Gantmacher. Cambridge University Press. Kenneth. Applied Numerical Linear Algebra.. M. ISBN 978-0521467131 • Lang. 2004). 1994). (October 16. Johnson. ISBN 978-0486671024 • Meyer. Springer. Topics in Matrix Analysis. (June 24.). ISBN 978-0387964126 • Marcus. Ljubic. Dover Publications. ISBN 978-0821826645 • Gelfand. (December 6. Undergraduate Texts in Mathematics. Springer. Paul R. ISBN 978-0898713893 • Gantmacher. ISBN 978-0898714548 • Mirsky. Springer. (1990). Charles R. (February 15. Dover Publications.). Marvin. The Linear Algebra a Beginning Graduate Student Ought to Know (2nd ed. Finite-Dimensional Vector Spaces. SIAM. Felix R. Van Loan. Springer. Dover Publications.). James W. Applications of the Theory of Matrices. I. Georgi E. Dover Publications.). (2005. Matrix Theory Vol.). linearalgebrawiki.edu/classes/linalg/linalg.math.ed. José Figueroa-O'Farrill.ca/linearalgebra/index.06/www) : MIT Course Website • MIT Linear Algebra Lectures (http://ocw.revoledu. 2000).ac.com/mathsym.edu/~bogacki/lat/).php?f=346): Interactive forums for discussion of linear algebra problems. (May 1. Cliffs Notes.edu/~math21b/) • Textbook and solutions manual (http://www. • Matrix and Linear Algebra Terms (http://www.org/book/) • Linear Algebra Wiki (http://www. ISBN 978-0071465793 • Zhang.html) with online interactive programs.numbertheory.mathlinks.com/topics/LinearAlgebra.technion. ISBN 978-0801891250 10 External links • International Linear Algebra Society (http://www.lamar.ac. McGraw-Hill.php) by Elmer G.tripod.edu/18. ISBN 978-0070380233 • McMahon. • Linear Algebra (http://mathworld. 3. Seymour (January 1. Fuzhen (April 7. Lamar University • Elementary Linear Algebra textbook with solutions (http://www.odu. . 2005). Marc (December 6.harvard. • Linear Algebra for Informatics (http://xmlearning.saylor. University of Edinburgh • Online Notes / Linear Algebra (http://tutorial. linear equations. The Johns Hopkins University Press. matrices.ro/Forum/index. Lipson.math.aspx) Paul Dawkins.org/courses/ma211/).courses.maths.mit.). 2009). Saylor Foundation.000 Solved Problems in Linear Algebra.il/iic/) • MIT Professor Gilbert Strang's Linear Algebra Course Homepage (http://web.html) and notation summary (http://planetmath. • Linear Algebra Solved Problems (http://www. Steven A.com/matrices.html) on PlanetMath. Schaum's Outline of Linear Algebra (3rd ed.wolfram.com/mathword.com/kardi/tutorial/LinearAlgebra/index.org/encyclopedia/LinearAlgebra. ISBN 978-0822053316 • Lipschutz. Wiens.tripod. 1996). Linear Algebra (Cliffs Quick Review).html) • Earliest Uses of Symbols for Matrices and Vectors (http://jeff560. McGraw–Hill Professional. from the lowest up to the hardest level (Putnam).Linear algebra Study guides and outlines • Leduc. • Linear Algebra overview (http://planetmath.org/) • Linear algebra (math 21b) homework and exercises (http://www. etc.uk/staff/aldrich/matrices. Linear Algebra: Challenging Problems for Students.html) on Earliest Uses of Various Mathematical Symbols (http://jeff560.math.uk).economics. ISBN 978-0071362009 • Lipschutz.html) on MathWorld.htm): free videos from MIT OpenCourseWare • Linear Algebra Toolkit (http://www.htm) on Earliest Known Uses of Some of the Words of Mathematics (http://jeff560. 1989).ac. Linear Algebra Demystified. Interactive web pages for vectors.html) • Linear Algebra (http://www.soton. Seymour.fas.egwald.org/encyclopedia/NotationInLinearAlgebra.mit. • Linear Algebra tutorial (http://people. David (October 28.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/ index. McGraw–Hill.tripod. html) Connell. Ruslan.numbertheory. Keith.ups.edu/linalg. Course of linear algebra and multidimensional geometry (http://arxiv.edu/~treil/papers/LADW/LADW. Rob.math. Edwin H.brown.edu/~ec/book/) Hefferon.html) .org/book/) Sharipov. Jim. A First Course in Linear Algebra (http://linear. Linear Algebra Done Wrong (http://www.html/) Matthews. Linear Algebra (http://joshua.Linear algebra 11 Online books • • • • • Beezer.math.HO/ 0405323) • Treil.edu/index. Sergei.smcvt.org/abs/math.miami.. Elementary Linear Algebra (http://www. Elements of Abstract and Linear Algebra (http://www. Arthur Rubin. Schzmo. Sanju dp. Zundark. Rossami. Wolfrock. Favonian. Jrdodge. Odedude. Loadedsalt. RedWolf. AxelBoldt. Dfrg. Gwaihir. Sklange. Quentin mcalmott. Dirac1933. Tbsmith. Cybercobra. Ivan Štambuk. Kaarebrandt.0 Contributors: Alksentrs at en.Article Sources and Contributors 12 Article Sources and Contributors Linear algebra Source: http://en. Zsniew. Discospinster. Cmdrjameson. Arcfrk.wikipedia. Licenses and Contributors File:Linear subspaces with shading. Hypercube. Bfigura. Gandalf61. David Eppstein. AvicAWB. GTBacchus. Egil. Heron. Fubar Obfusco. Gubbubu. Geometry guy. Seaphoto. Versus22. Daftman. Zginder. Ali'i. CSTAR. JohnBlackburne. Bryan Derksen. Blainster. OrenBochman. DVdm. GD. Jitse Niesen.org/w/index. Sarkar112. Redrose64. Revolver. Anakata. Natelewis.svg Source: http://en. Frecklefoot. Korg. Damian Yerrick. Romanm. Brion VIBBER. Sindrin7. 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