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

Kazuhisa Nakasho; Yuichi Futa; Hiroyuki Okazaki; Yasunari Shidama

Formalized Mathematics (2014)

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

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topKazuhisa 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 -

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