Linear Transformations of Euclidean Topological Spaces

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Linear Transformations of Euclidean Topological Spaces

We introduce linear transformations of Euclidean topological spaces given by a transformation matrix. Next, we prove selected properties and basic arithmetic operations on these linear transformations. Finally, we show that a linear transformation given by an invertible matrix is a homeomorphism.

[1] Jesse Alama. The rank+nullity theorem. Formalized Mathematics, 15(3):137-142, 2007, doi:10.2478/v10037-007-0015-6.

[2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 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, Mitsuru Aoki, Akio Matsumoto, and Yasunari Shidama. Processes in Petri nets. Formalized Mathematics, 11(1):125-132, 2003.

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

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

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

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

[10] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.

[11] Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.

[12] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.

[13] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23-28, 2003.

[14] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.

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

[16] Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181-187, 2005.

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

[18] Anna Lango and Grzegorz Bancerek. Product of families of groups and vector spaces. Formalized Mathematics, 3(2):235-240, 1992.

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

[20] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.

[21] Karol Pąk. Basic properties of determinants of square matrices over a field. Formalized Mathematics, 15(1):17-25, 2007, doi:10.2478/v10037-007-0003-x.

[22] Karol Pąk. Basic properties of the rank of matrices over a field. Formalized Mathematics, 15(4):199-211, 2007, doi:10.2478/v10037-007-0024-5.

[23] Karol Pąk. Solutions of linear equations. Formalized Mathematics, 16(1):81-90, 2008, doi:10.2478/v10037-008-0012-4.

[24] Karol Pąk. Linear map of matrices. Formalized Mathematics, 16(3):269-275, 2008, doi:10.2478/v10037-008-0032-0.

[25] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.

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

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

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

[29] Xiaopeng Yue, Xiquan Liang, and Zhongpin Sun. Some properties of some special matrices. Formalized Mathematics, 13(4):541-547, 2005.

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

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

Formalized Mathematics

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SCImago Journal Rank (SJR) 2017: 0.119
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researchers in the fields of formal methods and computer-checked mathematics

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