除环上的线性代数讲座
Lectures on Linear Algebra over Division Ring
摘要：
在这本书中，我处理线性代数上的除环。除环上的线性方程组具有与域上的线性方程组相似的性质。然而，产品的非交换性创造了一个新的画面。矩阵允许两个乘积通过转置连接。双环是在集合上定义环的两个相关结构的代数。在交换的情况下，一个线性方程组的解建立右或左向量空间，这取决于系统的类型。我们把向量空间和线性方程组一起研究，因为它们的性质有着密切的关系。与交换情形一样，向量空间的自同构群在标架流形上有一个传递表示。这给了我们一个介绍被动和主动表征的机会。研究除环上的向量空间，揭示了被动变换和主动变换之间关系的新细节，使这幅图更加清晰。考虑阿贝尔群中除环的孪生表示，得到了D向量空间及其线性映射的概念。在多加映射的基础上，给出了环的张量积和D向量空间的张量积的定义。
英文摘要：
In this book i treat linear algebra over division ring. A system of linear equations over a division ring has properties similar to properties of a system of linear equations over a field. However, noncommutativity of a product creates a new picture. Matrices allow two products linked by transpose. Biring is algebra which defines on the set two correlated structures of the ring. As in the commutative case, solutions of a system of linear equations build up right or left vector space depending on type of system. We study vector spaces together with the system of linear equations because their properties have a close relationship. As in a commutative case, the group of automorphisms of a vector space has a single transitive representation on a frame manifold. This gives us an opportunity to introduce passive and active representations. Studying a vector space over a division ring uncovers new details in the relationship between passive and active transformations, makes this picture clearer. Considering of twin representations of division ring in Abelian group leads to the concept of D vector space and their linear map. Based on polyadditive map I considered definition of tensor product of rings and tensor product of D vector spaces.
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