TAILIEUCHUNG - Ebook Abstract algebra - Theory and applications: Part 2
(BQ) Part 2 book "Abstract algebra - Theory and applications" has contents: Matrix groups and symmetry, group actions, the sylow theorems, rings, polynomials, integral domains, lattices and boolean algebras, lattices and boolean algebras, vector spaces, galois theory, hints and solutions, finite fields. | 12 Matrix Groups and Symmetry When Felix Klein (1849–1925) accepted a chair at the University of Erlangen, he outlined in his inaugural address a program to classify different geometries. Central to Klein’s program was the theory of groups: he considered geometry to be the study of properties that are left invariant under transformation groups. Groups, especially matrix groups, have now become important in the study of symmetry and have found applications in such disciplines as chemistry and physics. In the first part of this chapter, we will examine some of the classical matrix groups, such as the general linear group, the special linear group, and the orthogonal group. We will then use these matrix groups to investigate some of the ideas behind geometric symmetry. Matrix Groups Some Facts from Linear Algebra Before we study matrix groups, we must recall some basic facts from linear algebra. One of the most fundamental ideas of linear algebra is that of a linear transformation. A linear transformation or linear map T : Rn → Rm is a map that preserves vector addition and scalar multiplication; that is, for vectors x and y in Rn and a scalar α ∈ R, T (x + y) = T (x) + T (y) T (αy) = αT (y). An m × n matrix with entries in R represents a linear transformation from Rn to Rm . If we write vectors x = (x1 , . . . , xn )t and y = (y1 , . . . , yn )t in Rn 179 180 CHAPTER 12 MATRIX GROUPS AND SYMMETRY as column matrices, then an m × n matrix a11 a12 · · · a21 a22 · · · A= . . . . . . . am1 am2 · · · a1n a2n . . . amn maps the vectors to Rm linearly by matrix multiplication. Observe that if α is a real number, A(x + y) = Ax + Ay where and αAx = A(αx), x1 x2 x = . . . . xn We will often abbreviate the matrix A by writing (aij ). Conversely, if T : Rn → Rm is a linear map, we can associate a matrix A with T by considering what T does to the vectors e1 = (1, 0, . . . , 0)t e2 = (0, 1, . . . , 0)t . . . en = (0, 0, .
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