Linear Algebra and Geometry
This book on linear algebra and geometry is based on a course given by renowned academician I.R. Shafarevich at Moscow State University. The book begins with the theory of linear algebraic equations and the basic elements of matrix theory and continues wi
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Here we shall present a brief chronology of the appearance of the concepts discussed in this book. The development of mathematical ideas generally proceeds in such a way that some concepts gradually emerge from others. Therefore, it is generally impossible to fix accurately the appearance of some particular idea. We shall only point out the important milestones and, it goes without saying, shall do so only roughly. In particular, we shall limit our view to Western European mathematics. The principal stimulus was, of course, the creation of analytic geometry by Fermat and Descartes in the seventeenth century. This made it possible to specify points (on the line, in the plane, and in three-dimensional space) using numbers (one, two, or three), to specify curves and surfaces by equations, and to classify them according to the algebraic nature of their equations. In this regard, linear transformations were used frequently, especially by Euler, in the eighteenth century. Determinants (particularly as a symbolic apparatus for finding solutions of systems of n linear equations in n unknowns) were considered by Leibniz in the seventeenth century (even if only in a private letter) and in detail by Gabriel Cramer in the eighteenth. It is of interest that they were constructed on the basis of the rule of “general expansion” of the determinant, that is, on the basis of the most complex (among those that we considered in Chap. 2) way of defining them. This definition was discovered “empirically,” that is, conjectured on the basis of the formulas for the solution of systems of linear equations in two and three unknowns. The broadest use of determinants occurred in the nineteenth century, especially in the work of Cauchy and Jacobi. The concept of “multidimensionality,” that is, the passage from one, two, and three coordinates to an arbitrary number, was stimulated by the development of mechanics, where one considered systems with an arbitrary number of degrees of freedom. The idea of extending geometric intuition and concepts to this case was developed systematically by Cayley and Grassmann in the nineteenth century. At the same time, it became clear that one must study quadrics in spaces of arbitrary dimension (Jacobi and Sylvester in the nineteenth century). In fact, this question had already been considered by Euler. I.R. Shafarevich, A.O. Remizov, Linear Algebra and Geometry, DOI 10.1007/978-3-642-30994-6, © Springer-Verlag Berlin Heidelberg 2013
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Historical Note
The study of concepts defined by a set of abstract axioms (groups, rings, algebras, fields) began as early as the nineteenth century in the work of Hamilton and Cayley, but it reached its full flowering in the twentieth century, chiefly in the schools of Emmy Noether and Emil Artin. The concept of a projective space was first investigated by Desargues and Pascal in the seventeenth century, but systematic work in this direction began only in the nineteenth century, beginning with the work of Poncelet. The axiomatic definition of vector spaces and Euclidea
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