Lectures In Geometry: Linear Algebra and Differential Geometry (semester Ii)

Argumento de Lectures In Geometry: Linear Algebra and Differential Geometry (semester Ii)

This book is a direct continuation of the author's previous book* and is akin to it in being a nearly faithful record of the Lectures delivered by the author in the second semester of the first, year at the Mathematics-Mechanics Faculty of Moscow State University named after M. V. Lomonosov to mathematical students (a course in Linear Algebra and Analytic Geometry). Naturally, in the selection of the material and the order of presentation the author was guided by the same considerations as in the first semester (see the Preface in [1]). The number of Lectures in the book is explained by the fact that although the curriculum assigns 32 Lectures to the course, in practice it is impossible to deliver more than 27 Lectures.

The course in Linear Algebra and Analytic Geometry is just a part of a single two-year course in geometry, and much in this book is accounted for, as regards the choice of the material and its accentuation, by orientation to the second year devoted to the differential geometry of manifolds. In particular, it has proved possible (although it is not envisaged by the curriculum) to transfer part of the propaedeutic material of the third semester (the elementary differential geometry of curves and surfaces in three-dimensional space) to the second-semester course and this has substantially facilitated (not only for the Lecturer but, what is of course more important, also for the students) the third semester course. At the same time, as experience has shown, this material appeals to the students and they learn it well on the whole already in the second semester. M. M. Postnikov October 27, 19770Preface

Lecture 1
Vector spaces. Subspaces. Intersection of subspaces. Linear spans. A sum of subspaces. The dimension of a subspace. The dimension of a sum of subspaces. The dimension of a linear span.

Lecture 2
Matrix rank theorem. The rank of a matrix product. The Kronecker-Capelli theorem. Solution of systems of linear equations.

Lecture 3
Direct sums of subspaces. Decomposition of a space as a direct sum of subspacea. Factor spaces. Homomorphisms of vector spaces. Direct sums of spaces.

Lecture 4
The conjugate space. Dual spaces. A second conjugate space. The transformation of a conjugate basis and of the coordinates of covectors. Annulets. The space of solutions of a system of homogeneous linear equations.

Lecture 5
An annulet of an annulet and annulets of direct summands. Bilinear functionals and bilinear forms. Bilinear func-tionals in a conjugate space. Mixed bilinear functionals. Tensors.

Lecture 6
Multiplication of tensors. The basis of a space of tensors. Contraction of tensors. The rank space of a multilinear functional.

Lecture 7
The rank of a multilinear functional. Functionals and permutations. Alternation.

Lecture 8
Skew-symmetric multilinear functionals. External multiplication. Grassman algebra. External sums of covectors. Expansion of skew-symmetric functionals with respect to the external products of covectors of a basis.

Lecture 9
The basis of a space of skew-symmetric functionals. Formulas for the transformation of the basis of that space. Multivec-tors...The external rank of a skew-symmetric functional. Multi vector rank theorem. Conditions for the equality of multivectors.

Lecture 10
Cartan/s divisibility theorem. Pliicker relations. The Plu-cker coordinates of subspaces. Planes in an affine space. Planes in a projective space and their coordinates.

Lecture 11
Symmetric and skew-symmetric bilinear functionals. A matrix of symmetric bilinear functionals. The rank of a bilinear functional. Quadratic functionals and quadratic forms. Lagrange theorem.

Lecture 12
Jacobi theorem. Quadratic forms over the fields of complex and real numbers. The law of inertia. Positively definite quadratic functionals and forms.

Lecture 13
Second degree hypersurfaces of an n-dimensional projective space. Second degree hypersurfaces in a complex and a real-complex projective space. Second degree hypersurfaces of an n-dimensional affine space. Second degree nypersurfaces in a complex and a real-complex affine space.

Lecture 14
The algebra of linear operators. Operators and mixed bilinear functionals. Linear operators and matrices. Invertible operators. The adjoint operator. The Fredholm alternative. Invariant subspaces and induced operators.

Lecture 15
Eigenvalues. Characteristic roots. Diagonalizable operators. Operators with simple spectrum. The existence of a basis in which the matrix of an operator is triangular. Nilpo-tent operators.

Lecture 16
Decomposition of a nilpotent operator as a direct sum of cyclic operators. Root subspaces. Normal Jordan form. The Hamilton-Cayley theorem.

Lecture 17
Complexificationof a linear opera tor. Proper subspaces belonging to characteristic roots. Operators whose complexifica-tion is diagonalizable.

Lecture 18
Euclidean and unitary spaces. Orthogonal complements. The identification of vectors and covectors. Annulets and orthogonal complements. Bilinear functionals and linear operators. Elimination of arbitrariness in the identification of tensors of different types. The metric tensor. Lowering and raising of indices.

Lecture 19
Adjoint operators. Self-adjoint operators. Skew-symmetric and skew-Hermitian operators. Analogy between Hermitian operators and real numbers. Spectral properties of self-adjoint operators. The orthogonal diagonalizability of self-adjoint operators.

Lecture 20
Bringing quadratic forms into canonical form by orthogonal transformation of variables. Second degree hypersurfaces in a Euclidean point space. The minimax property of eigenvalues of self-adjoint operators. Orthogonally diagonalizable operators.

Lecture 21
Positive operators. Isometric operators. Unitary matrices. Polar factorization of invertible operators. A geometrical interpretation of polar factorization. Parallel translations and centroaffine transformations. Bringing a unitary operator into diagonal form. A rotation of an ^-dimensional Euclidean space as a composition of rotations in two-dimensional planes.

Lecture 22
Smooth functions. Smooth hypersurfaces. Gradient. Derivatives with respect to a vector. Vector fields. Singular points of a vector field. A module of vector fields. Potential and ir-rotational vector fields. The rotation of a vector field. The divergence of a vector field. Vector analysis. Hamilton's symbolic vector. Formulas for products. Compositions of operators.

Lecture 23
Continuous, smooth, and regular curves. Equivalent curves. Regular curves in the plane and graphs of functions. The tangential hyperplane of a 'hypersurface. The length of a curve. Curves in the plane. Curves in a three-dimensional space.

Lecture 24
Projections of a curve onto the coordinate planes of the moving n-hedron. Prenet's formulas for a curve in the n-dimen-sional space. Representation of a curve by its curvatures. Regular surfaces. Examples of surfaces.

Lecture 25
Vectors tangential to a surface. The tangential plane. The first quadratic form of a surface. Mensuration of lengths and angles on a surface. Diffeomorphismsof surfaces. Isometries and the intrinsic geometry of a surface. Examples. Developables.

Lecture 26
The tangential plane and the normal vector. The curvature of a normal section. The second quadratic form of a surface. The indicatfix of Dupin. Principal curvatures. The second quadratic form of a graph. Ruled surfaces of zero curvature. Surfaces of revolution.

Lecture 27
Weingarten's derivation formulas. Coefficients of connection. The Gauss theorem. The necessary and sufficient conditions of isometry.

Subject Index.