高等代数II（实验班）

北京大学数学科学学院

This is the second half of a one-year honors course on freshmen algebra at Peking University. It is designed to help talented students to enter various fields of modern mathematics earlier. Comparing with the traditional course "Advanced Algebra", this course aims at a higher level in its depth, width and abstraction. The central topic is an axiomatic treatment of linear spaces and linear maps. Matrix algebra is developed as a tool for studying abstract concepts. Basic notion of tensor algebra, groups, rings, modules and fields is also introduced. In the second half, we will concentrate on more properties of linear maps, canonical forms of matrices and linear transformations, inner product spaces, bilinear functions, and quadratic forms.

Linear Algebra,K. Hoffman and R. Kunze,世界图书出版公司,2008,2；


一、        线性映射II（14学时）
特征值和特征向量、特征多项式、极小多项式、Cayley-Hamilton定理、线性变换与矩阵的对角化、不变子空间、特征子空间和广义特征子空间、空间的准素分解。
二、        线性变换和矩阵的标准形（16学时）
环上的模、线性变换诱导的模结构、零化多项式、循环子空间和循环分解、Jordan标准形、有理标准形、幂零与半单变换、线性变换的Jordan分解。
三、        内积空间（14学时）
内积的概念、Gram-Schmidt正交化、子空间的正交补、正交群与酉群、线性变换的伴随变换、对称变换与Hermite变换的对角化和谱分解、正定变换、极分解。
四、        双线性函数与二次型（10学时）
双线性函数的矩阵、矩阵的合同、对称双线性函数与二次型、二次型的分类、正定性、反对称双线性函数与辛群。
课堂讲授
作业10%，期中考试30%，期末考试60%