Chapter 1 deals with systems of linear equations and their solution by means of elementary row operations on matrices. It has been our practice to spend about six lectures on this material. It provides the student with some picture of the origins of linear algebra and with the computational technique necessary to understand examples of the more abstract ideas occurring in the later chapters. Chapter 2 deals with vector spaces， subspaces， bases， and dimension. Chapter 3 treats linear transformations， their algebra， their representation by matrices， as well as isomorphism， linear functionals， and dual spaces. Chapter 4 defines the algebra of polynomials over a field， the ideals in that algebra， and the prime factorization of a polynomial. It also deals with roots， Taylor's formula， and the Lagrange interpolation formula. Chapter 5 develops determinants of square matrices， the determinant being viewed as an alternating n-linear function of the rows of a matrix， and then proceeds to multilinear functions on modules as well as the Graesman ring. The material on modules places the concept of determinant in a wider and more comprehensive setting than is usually found in elementary textbooks. Chapters 6 and 7 contain a discussion of the concepts which are basic to the analysis of a single linear transformation on a finite-dimensional vector space; the analysis of charac-teristic （eigen） values， triangulable and diagonalizable transformations; the concepte of the diagonalizable and nilpotent parts of a more general transformation， and the rational and Jordan canonical forms. The primary and cyclic decomposition theorems play a central role， the latter being arrived at through the study of admissible subspaces. Chapter 7 includes a discussion of matrices over a polynomial domain， the computation of invariant factors and elementary divisors of a matrix， and the development of the Smith canonical form. The chapter ends with a discuseion of semi-simple operators， to round out the analysis of a single operator. Chapter 8 treats finite-dimensional inner product spaces in some detail. It covers the basic geometry， relating orthogonalization to the idea of 'best approximation to a vector' and leading to the concepts of the orthogonal projection of a vector onto a subspace and the orthogonal complement of a subspace. The chapter treats unitary operators and culminates in the diagonalization of seff-adjoint and normal operators. Chapter 9 introduces sesqui-linear forms， relates them to positive and seff-adjoint operators on an inner product space， moves on to the spectral theory of normal operators and then to more sophisticated results concerning normal operators on real or complex inner product spaces. Chapter 10 discusses bilinear forms， emphasizing canonical forms for symmetric and skew-symmetric forms， as well as groups preserving non-degenerate forms， especially the orthogonal， unitary， pseudo-orthogonal and Lorentz groups.
鄂公网安备 42010302001612号 