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

Formalized Mathematics (2014)

• Volume: 22, Issue: 3, page 189-198
• ISSN: 1426-2630

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## Abstract

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In this article, we formalize some basic facts of Z-module. In the first section, we discuss the rank of submodule of Z-module and its properties. Especially, we formally prove that the rank of any Z-module is equal to or more than that of its submodules, and vice versa, and that there exists a submodule with any given rank that satisfies the above condition. In the next section, we mention basic facts of linear transformations between two Z-modules. In this section, we define homomorphism between two Z-modules and deal with kernel and image of homomorphism. In the last section, we formally prove some basic facts about linearly independent subsets and linear combinations. These formalizations are based on [9](p.191-242), [23](p.117-172) and [2](p.17-35).

## How to cite

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Kazuhisa Nakasho, et al. "Rank of Submodule, Linear Transformations and Linearly Independent Subsets of Z-module." Formalized Mathematics 22.3 (2014): 189-198. <http://eudml.org/doc/270942>.

@article{KazuhisaNakasho2014,
abstract = {In this article, we formalize some basic facts of Z-module. In the first section, we discuss the rank of submodule of Z-module and its properties. Especially, we formally prove that the rank of any Z-module is equal to or more than that of its submodules, and vice versa, and that there exists a submodule with any given rank that satisfies the above condition. In the next section, we mention basic facts of linear transformations between two Z-modules. In this section, we define homomorphism between two Z-modules and deal with kernel and image of homomorphism. In the last section, we formally prove some basic facts about linearly independent subsets and linear combinations. These formalizations are based on [9](p.191-242), [23](p.117-172) and [2](p.17-35).},
author = {Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {free Z-module; rank of Z-module; homomorphism of Z-module; linearly independent; linear combination; free -module; rank of -module; homomorphism of -module},
language = {eng},
number = {3},
pages = {189-198},
title = {Rank of Submodule, Linear Transformations and Linearly Independent Subsets of Z-module},
url = {http://eudml.org/doc/270942},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Kazuhisa Nakasho
AU - Yuichi Futa
AU - Hiroyuki Okazaki
AU - Yasunari Shidama
TI - Rank of Submodule, Linear Transformations and Linearly Independent Subsets of Z-module
JO - Formalized Mathematics
PY - 2014
VL - 22
IS - 3
SP - 189
EP - 198
AB - In this article, we formalize some basic facts of Z-module. In the first section, we discuss the rank of submodule of Z-module and its properties. Especially, we formally prove that the rank of any Z-module is equal to or more than that of its submodules, and vice versa, and that there exists a submodule with any given rank that satisfies the above condition. In the next section, we mention basic facts of linear transformations between two Z-modules. In this section, we define homomorphism between two Z-modules and deal with kernel and image of homomorphism. In the last section, we formally prove some basic facts about linearly independent subsets and linear combinations. These formalizations are based on [9](p.191-242), [23](p.117-172) and [2](p.17-35).
LA - eng
KW - free Z-module; rank of Z-module; homomorphism of Z-module; linearly independent; linear combination; free -module; rank of -module; homomorphism of -module
UR - http://eudml.org/doc/270942
ER -

## References

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1. [1] Jesse Alama. The rank+nullity theorem. Formalized Mathematics, 15(3):137-142, 2007. doi:10.2478/v10037-007-0015-6.[Crossref]
2. [2] Michael Francis Atiyah and Ian Grant Macdonald. Introduction to Commutative Algebra, volume 2. Addison-Wesley Reading, 1969. Zbl0175.03601
3. [3] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
4. [4] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.
5. [5] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
6. [6] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
7. [7] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
8. [8] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.
9. [9] Nicolas Bourbaki. Elements of Mathematics. Algebra I. Chapters 1-3. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1989. Zbl0673.00001
10. [10] Czesław Bylinski. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990.
11. [11] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.
12. [12] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
13. [13] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
14. [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.
15. [15] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
16. [16] Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.
17. [17] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
18. [18] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
19. [19] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Z-modules. Formalized Mathematics, 20(1):47-59, 2012. doi:10.2478/v10037-012-0007-z.[Crossref] Zbl1276.94012
20. [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.[Crossref] Zbl06213839
21. [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.[Crossref] Zbl06213848
22. [22] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
23. [23] Serge Lang. Algebra. Springer, 3rd edition, 2005.
24. [24] Dariusz Surowik. Cyclic groups and some of their properties - part I. Formalized Mathematics, 2(5):623-627, 1991.
25. [25] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.
26. [26] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.
27. [27] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.
28. [28] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
29. [29] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
30. [30] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1 (5):877-882, 1990.
31. [31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
32. [32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
33. [33] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

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