Rank of Submodule, Linear Transformations and Linearly Independent Subsets of Z-module

[1] Jesse Alama. The rank+nullity theorem. Formalized Mathematics, 15(3):137-142, 2007. doi:10.2478/v10037-007-0015-6.10.2478/v10037-007-0015-6Search in Google Scholar

[2] Michael Francis Atiyah and Ian Grant Macdonald. Introduction to Commutative Algebra, volume 2. Addison-Wesley Reading, 1969.Search in Google Scholar

[3] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Search in Google Scholar

[4] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.Search in Google Scholar

[5] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Search in Google Scholar

[6] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Search in Google Scholar

[7] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Search in Google Scholar

[8] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Search in Google Scholar

[9] Nicolas Bourbaki. Elements of Mathematics. Algebra I. Chapters 1-3. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1989.Search in Google Scholar

[10] Czesław Bylinski. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990.Search in Google Scholar

[11] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Search in Google Scholar

[12] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Search in Google Scholar

[13] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Search in Google Scholar

[14] Czesław Bylinski. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.Search in Google Scholar

[15] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Search in Google Scholar

[16] Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Search in Google Scholar

[17] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Search in Google Scholar

[18] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Search in Google Scholar

[19] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Z-modules. Formalized Mathematics, 20(1):47-59, 2012. doi:10.2478/v10037-012-0007-z.10.2478/v10037-012-0007-zSearch in Google Scholar

[20] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of Z-module. Formalized Mathematics, 20(3):205-214, 2012. doi:10.2478/v10037-012-0024-y.10.2478/v10037-012-0024-ySearch in Google Scholar

[21] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Free Z-module. Formalized Mathematics, 20(4):275-280, 2012. doi:10.2478/v10037-012-0033-x.10.2478/v10037-012-0033-xSearch in Google Scholar

[22] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Search in Google Scholar

[23] Serge Lang. Algebra. Springer, 3rd edition, 2005.Search in Google Scholar

[24] Dariusz Surowik. Cyclic groups and some of their properties - part I. Formalized Mathematics, 2(5):623-627, 1991.Search in Google Scholar

[25] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.Search in Google Scholar

[26] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Search in Google Scholar

[27] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Search in Google Scholar

[28] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Search in Google Scholar

[29] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Search in Google Scholar

[30] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1 (5):877-882, 1990.Search in Google Scholar

[31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Search in Google Scholar

[32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Search in Google Scholar

[33] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990. Search in Google Scholar