81-商空间及其应用（81- quotient spaces and their applications）.docVIP

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81-商空间及其应用（81- quotient spaces and their applications）.doc

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81-商空间及其应用（81- quotient spaces and their applications）

81-商空间及其应用（81- quotient spaces and their applications） The quotient space and its application Yong Lin Liu (Department of mathematics, Fujian Wuyi University Wuyishan 354300) Abstract: the properties of quotient space, gives the basic theorem on quotient space, the first second isomorphism theorem and homomorphism, As an application, proved the formula two famous dimension in linear algebra is a direct consequence of them. Keywords: linear algebra, quotient space, isomorphism theorem, dimension formula CLC number: O151.2 MR (2000): 15A03 Quotient Spaces and Applications LIU Yonglin (Mathematics Department of Wuyi University, Wuyishan, Fujian 354300) Abstract: This paper investigates the properties of quotient spaces and obtains the first, second isomorphism theorems and homomorphism fundamental theorem. As the applications, we prove that two famous dimensional formulas in linear algebras are their corollaries. Key words:Linear algebra, quotient space, isomorphism theorem, dimensional formula 1. concepts of quotient space In modern algebra, the concept of business of the group theory, the concept of taking the ring theory. Similarly, the concept in the linear space theory we can quotient space. Let W be a linear space V number domain P subspace, using W to define a relationship between two yuan in V vector: ~ IFF - W Proposition 1.1 ~ is a congruence relation of V. Proof: in V, a = 0 in W, so, ~; in V, if ~ - = 0, W, and - (-) - W in W, that is, therefore, ~;,, gamma in V, and if ~ ~ gamma, gamma -, and W so, gamma = (-) + (gamma) = W ~ gamma ~, so, therefore is an equivalence relation. ,, ETA, zeta = W, if ~ ~, ETA - ETA, epsilon, zeta, W, and (+ n) (+ x) = (-) + (ETA zeta) = W, and + ETA zeta ~ +. So ~ is a congruence relation. So, V = {x | ~ x} vector equivalence, easy to prove = + W = {+ | in W. We can call these equivalence class subspace to W with the sale, because in a linear space vector addition on commutative, it is not necessary to distinguish between left