Contents. History From the study of determinants and matrices to modern linear algebra The study of linear algebra first emerged from the introduction of, for solving.

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Determinants were considered by in 1693, and subsequently, in 1750, used them for giving explicit solutions of linear system, now called. Later, further developed the theory of solving linear systems by using, which was initially listed as an advancement in. The study of matrix algebra first emerged in England in the mid-1800s.

In 1844 published his 'Theory of Extension' which included foundational new topics of what is today called linear algebra. In 1848, introduced the term matrix, which is Latin for 'womb'.

While studying compositions of linear transformations, 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'. In 1882, wrote the book titled 'Linear Algebra'. The first modern and more precise definition of a vector space was introduced by in 1888; 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.

The use of matrices in, and helped spread the subject of linear algebra beyond pure mathematics. The development of computers led to increased research in efficient for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations. The origin of many of these ideas is discussed in the articles on and. Educational history Linear algebra first appeared in American graduate textbooks in the 1940s and in undergraduate textbooks in the 1950s. Following work by the, U.S. High schools asked 12th grade students to do 'matrix algebra, formerly reserved for college' in the 1960s. In France during the 1960s, educators attempted to teach linear algebra through finite-dimensional vector spaces in the first year of.

This was met with a backlash in the 1980s that removed linear algebra from the curriculum. In 1993, the U.S.-based Linear Algebra Curriculum Study Group recommended that undergraduate linear algebra courses be given an application-based 'matrix orientation' as opposed to a theoretical orientation. Reviews of the teaching of linear algebra call for stress on visualization and geometric interpretation of theoretical ideas, and to include the jewel in the crown of linear algebra, the (SVD), as 'so many other disciplines use it'.

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To better suit 21st century applications, such as data mining and uncertainty analysis, linear algebra can be based upon the SVD instead of Gaussian Elimination. Scope of study Vector spaces. Main article: The main structures of linear algebra are vector spaces. A vector space over a F (often the field of the ) is a V equipped with two satisfying the following axioms. Elements of V are called vectors, and of F are called scalars. The first operation, 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 av.

The operations of addition and multiplication in a vector space must satisfy the following. In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars in F. Axiom Signification of addition u + ( v + w) = ( u + v) + w of addition u + v = v + u of addition There exists an element 0 ∈ V, called the, such that v + 0 = v for all v ∈ V. Of addition For every v ∈ V, there exists an element − v ∈ V, called the of v, such that v + (− v) = 0 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 Identity element of scalar multiplication 1 v = v, where 1 denotes the in F. The first four axioms are those of V being an under vector addition. Elements of a vector space may have various nature; for example, they can be,.

Linear algebra is concerned with properties common to all vector spaces. Linear transformations. Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC,. Strang, Gilbert (July 19, 2005), Linear Algebra and Its Applications (4th ed.), Brooks Cole,. Weisstein, Eric.

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Archived from on 2012-09-10. Retrieved 2014-07-08. (1993). 'The Growing Importance of Linear Algebra in Undergraduate Mathematics'.

College Mathematics Journal. 24 (1): 3–9. Goodlad, John I.; von stoephasius, Reneta; Klein, M. Frances (1966). Department of Health, Education, and Welfare: Office of Education.

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Retrieved 9 July 2014. Dorier, Jean-Luc; Robert, Aline; Robinet, Jacqueline; Rogalsiu, Marc (2000). Dorier, Jean-Luc, ed. Retrieved 9 July 2014. Carlson, David; Johnson, Charles R.; Lay, David C.; Porter, A. Duane (1993). 'The Linear Algebra Curriculum Study Group Recommendations for the First Course in Linear Algebra'.

The College Mathematics Journal. 24 (1): 41–46. Schumacher, Martha J. Siegel, and Paul Zorn (2015) 2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences. The Mathematical Association of America. Turner et al.

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(2015) Modeling across the Curriculum II. Report on the second SIAM-NSF Workshop, Alexandria, VA. Roberts (2017) Linear Algebra Reformed for 21st-C Application. 27. Axler (2004), pp. 28–29. The existence of a basis is straightforward for vector spaces, and for vector spaces, but in it is to the.

Axler (2204), p. 33. Axler (2004), p.

55. If we restrict to integers, then only 1 and -1 have an inverse. Consequently, the inverse of an integer matrix is an integer matrix if and only if the determinant is 1 or -1. Jain, Khalil Ahmad (1995). '5.1 Definitions and basic properties of inner product spaces and Hilbert spaces'.

New Age International. Eduard Prugovec̆ki (1981). 'Definition 2.1'. Academic Press.

Gunawardena, Jeremy. Harvard Medical School. Retrieved 2 May 2012. Miller, Steven. Brown University. Retrieved 1 May 2013. ^ Strang, Gilbert (July 19, 2005), Linear Algebra and Its Applications (4th ed.), Brooks Cole,.

^ J. Semple and G. Kneebone, Algebraic Projective Geometry, Clarendon Press, London, 1952. Nering, Linear Algebra and Matrix Theory, John-Wiley, New York, NY, 1963.