## Summary

In this article, we formalize a matrix of ℤ-module and its properties. Specially, we formalize a matrix of a linear transformation of ℤ-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free ℤ-module *V*, determinant of its Gramian matrix is constant regardless of selection of its basis. ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [22] and coding theory [14]. Some theorems in this article are described by translating theorems in [24], [26] and [19] into theorems of ℤ-module.

^{1}This work was supported by JSPS KAKENHI 21240001 and 22300285.

## References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3): 537–541, 1990.

[3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[6] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[7] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643–649, 1990.

[8] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.

[10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[14] Wolfgang Ebeling. Lattices and Codes. Advanced Lectures in Mathematics. Springer Fachmedien Wiesbaden, 2013.

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

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

[17] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2 (4):475–480, 1991.

[18] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.

[19] Jarosław Kotowicz. Bilinear functionals in vector spaces. Formalized Mathematics, 11(1): 69–86, 2003.

[20] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.

[21] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[22] Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: a cryptographic perspective. The International Series in Engineering and Computer Science, 2002.

[23] Anna Justyna Milewska. The Hahn Banach theorem in the vector space over the field of complex numbers. Formalized Mathematics, 9(2):363–371, 2001.

[24] Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339–345, 1996.

[25] Michał Muzalewski. Rings and modules – part II. Formalized Mathematics, 2(4):579–585, 1991.

[26] Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29–34, 1993.

[27] Karol Pąk and Andrzej Trybulec. Laplace expansion. Formalized Mathematics, 15(3): 143–150, 2007. doi:10.2478/v10037-007-0016-5.

[28] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29–34, 1999.

[29] Nobuyuki Tamura and Yatsuka Nakamura. Determinant and inverse of matrices of real elements. Formalized Mathematics, 15(3):127–136, 2007. doi:10.2478/v10037-007-0014-7.

[30] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.

[31] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

[32] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.

[33] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

[34] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.

[35] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[36] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1(5):877–882, 1990.

[37] Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883–885, 1990.

[38] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[39] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.

[40] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

[41] Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205–211, 1992.

[42] Katarzyna Zawadzka. The product and the determinant of matrices with entries in a field. Formalized Mathematics, 4(1):1–8, 1993.