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

• Autorzy: Kazuhisa Nakasho , Yuichi Futa , Hiroyuki Okazaki , Yasunari Shidama
• Języki publikacji: EN

Abstrakty

EN

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).

Słowa kluczowe

EN

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

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2014
• Tom: 22
• Numer: 3
• Strony: 189-198

Daty

wydano

2014-09-01

otrzymano

2014-07-10

online

2015-04-30

Twórcy

autor

Kazuhisa Nakasho

• Shinshu University Nagano, Japan

autor

Yuichi Futa

• Japan Advanced Institute of Science and Technology Ishikawa, Japan

autor

Hiroyuki Okazaki

• Shinshu University Nagano, Japan

autor

Yasunari Shidama

• Shinshu University Nagano, Japan

Bibliografia

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Identyfikatory

DOI

10.2478/forma-2014-0021