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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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1898-9934
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Volume 22, Issue 3

Issues

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

Kazuhisa Nakasho / Yuichi Futa / Hiroyuki Okazaki / Yasunari Shidama
Published Online: 2014-03-31 | DOI: https://doi.org/10.2478/forma-2014-0021

Summary

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).

Keywords : free Z-module; rank of Z-module; homomorphism of Z-module; linearly independent; linear combination

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About the article

Received: 2014-07-10

Published Online: 2014-03-31

Published in Print: 2014-09-01


Citation Information: Formalized Mathematics, Volume 22, Issue 3, Pages 189–198, ISSN (Online) 1898-9934, DOI: https://doi.org/10.2478/forma-2014-0021.

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© by Kazuhisa Nakasho. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0) BY-SA 3.0

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