文章主要构建了置换群群元素对应的置换矩阵在复数域上全矩阵代数的中心化子代数的胞腔基,从而说明置换群群元素对应的置换矩阵在复数域上全矩阵代数的中心化子代数是胞腔代数。文章的证明根据章节分成三部分内容,第一部分给出了胞腔代...文章主要构建了置换群群元素对应的置换矩阵在复数域上全矩阵代数的中心化子代数的胞腔基,从而说明置换群群元素对应的置换矩阵在复数域上全矩阵代数的中心化子代数是胞腔代数。文章的证明根据章节分成三部分内容,第一部分给出了胞腔代数的定义,中心化子代数的定义,以及一些符号的意义。然后第二部分根据循环群在复数域上的群代数和某些置换矩阵的中心化子代数的联系,证明了循环群在复数域上的群代数是胞腔的。文章的第三部分是根据置换群的群元素可以表示成不相交轮换的乘积,通过其置换矩阵的分块乘积,发现该中心化子的结构和循环群在复数域上的基的联系,最后根据循环群在复数域上的胞腔基构造了一般的置换矩阵在复数域上全矩阵代数的中心化子代数的胞腔基。The article mainly constructs the cellular basis of the centralizer of the permutation matrix corresponding to the elements of the permutation group in the full matrix algebra over the complex field, thus showing that the centralizer of the permutation matrix corresponding to the elements of the permutation group in the full matrix algebra over the complex field is a cellular algebra. The proof of the article is divided into three parts according to the sections. The first part gives the definition of cellular algebra, the definition of centralizer algebra, and the meaning of some symbols. Then, the second part proves that the group algebra of the cyclic group over the complex field is cellular based on the connection between the group algebra of the cyclic group over the complex field and the centralizer algebra of some permutation matrices. The third part of the article discovers the connection between the structure of the centralizer and the basis of the cyclic group over the complex field through the block product of the permutation matrix according to the fact that the group elements of the permutation group can be expressed as the product of disjoint cycles, and then constructs the cellular basis of the centralizer of the general permutation matrix in the full matrix algebra over the complex field based on the cellular basis of the cyclic group over the complex field.展开更多
文章首先从四个方面提出了新工科背景下线性代数课程思政的挖掘方式与教学目的,即教师在传授知识的过程中通过线性代数包含的数学文化故事、与之相关的生活案例、科学实践以及课程本身所蕴含的传统文化和哲学道理来培养学生的家国情怀...文章首先从四个方面提出了新工科背景下线性代数课程思政的挖掘方式与教学目的,即教师在传授知识的过程中通过线性代数包含的数学文化故事、与之相关的生活案例、科学实践以及课程本身所蕴含的传统文化和哲学道理来培养学生的家国情怀、爱国主义、科研精神以及唯物主义辩证思想。其次,探索并给出了线性代数教学中课程思政的实践路径,从而为教师在新工科背景下线性代数教学中融入课程思政提供一定的参考。This paper first proposes the exploration methods and teaching objectives of courses ideological and political education from four aspects in linear algebra under the background of new engineering, that is, during the process of imparting knowledge, teachers should cultivate students’ patriotism, scientific research spirit, and materialistic and dialectic philosophy through traditional culture and philosophical principles contained in mathematical cultural stories, related life cases, scientific practices, and the curriculum itself of linear algebra. Secondly, the practical paths of integrating ideological and political education into linear algebra teaching have been explored and provided, thus it provides some reference for teachers to integrate ideological and political education into linear algebra teaching in the context of new engineering.展开更多
文摘文章主要构建了置换群群元素对应的置换矩阵在复数域上全矩阵代数的中心化子代数的胞腔基,从而说明置换群群元素对应的置换矩阵在复数域上全矩阵代数的中心化子代数是胞腔代数。文章的证明根据章节分成三部分内容,第一部分给出了胞腔代数的定义,中心化子代数的定义,以及一些符号的意义。然后第二部分根据循环群在复数域上的群代数和某些置换矩阵的中心化子代数的联系,证明了循环群在复数域上的群代数是胞腔的。文章的第三部分是根据置换群的群元素可以表示成不相交轮换的乘积,通过其置换矩阵的分块乘积,发现该中心化子的结构和循环群在复数域上的基的联系,最后根据循环群在复数域上的胞腔基构造了一般的置换矩阵在复数域上全矩阵代数的中心化子代数的胞腔基。The article mainly constructs the cellular basis of the centralizer of the permutation matrix corresponding to the elements of the permutation group in the full matrix algebra over the complex field, thus showing that the centralizer of the permutation matrix corresponding to the elements of the permutation group in the full matrix algebra over the complex field is a cellular algebra. The proof of the article is divided into three parts according to the sections. The first part gives the definition of cellular algebra, the definition of centralizer algebra, and the meaning of some symbols. Then, the second part proves that the group algebra of the cyclic group over the complex field is cellular based on the connection between the group algebra of the cyclic group over the complex field and the centralizer algebra of some permutation matrices. The third part of the article discovers the connection between the structure of the centralizer and the basis of the cyclic group over the complex field through the block product of the permutation matrix according to the fact that the group elements of the permutation group can be expressed as the product of disjoint cycles, and then constructs the cellular basis of the centralizer of the general permutation matrix in the full matrix algebra over the complex field based on the cellular basis of the cyclic group over the complex field.
文摘文章首先从四个方面提出了新工科背景下线性代数课程思政的挖掘方式与教学目的,即教师在传授知识的过程中通过线性代数包含的数学文化故事、与之相关的生活案例、科学实践以及课程本身所蕴含的传统文化和哲学道理来培养学生的家国情怀、爱国主义、科研精神以及唯物主义辩证思想。其次,探索并给出了线性代数教学中课程思政的实践路径,从而为教师在新工科背景下线性代数教学中融入课程思政提供一定的参考。This paper first proposes the exploration methods and teaching objectives of courses ideological and political education from four aspects in linear algebra under the background of new engineering, that is, during the process of imparting knowledge, teachers should cultivate students’ patriotism, scientific research spirit, and materialistic and dialectic philosophy through traditional culture and philosophical principles contained in mathematical cultural stories, related life cases, scientific practices, and the curriculum itself of linear algebra. Secondly, the practical paths of integrating ideological and political education into linear algebra teaching have been explored and provided, thus it provides some reference for teachers to integrate ideological and political education into linear algebra teaching in the context of new engineering.
