Linear Algebra and Geometry

Abstract

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.

Infinitesimal Lyapunov functions for singular flows

“…, v n } is orthonormal. This can be done smoothly in a neighborhood of x in M since we are assuming that the quadratic forms are non-degenerate; the reader can check the method of Lagrange in a standard Linear Algebra textbook and observe that the steps can be performed robustly and smoothly for all nearby tangent spaces; see for instance [41,33] and Example 6 below. In this adapted inner product we have that J x has entries from {−1, 0, 1} only, J * x = J x and also that J 2 x = J x .…”

Cauchy–Binet for pseudo-determinants

“…This lemma appears as Theorem A.2.1) in [21]. As pointed out in [22] Lemma 10.16), it rephrases the classical matrix multiplication of the two exterior products Λ k A, Λ k B as a composition of maps Λ k R n → Λ k R m → Λ k R n . For more on the exterior algebra, see [1,20].…”

The Visualization of Spherical Patterns with Symmetries of the Wallpaper Group

“…Or, equivalently, for any basis vectors of 2 , the subgroup of all linear combinations with integer coefficients of the vectors forms a lattice [12,13]. Since a lattice is a finitely generated free abelian group, it is isomorphic to 2 and fully spans the real vector space 2 [14]. A lattice may be viewed as a regular tiling of a space by a primitive cell.…”