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Linear Map of Matrices

Formalized Mathematics

Volume 16 (2008): Issue 3 (September 2008)

By: Karol Pąk

Open Access

|Mar 2009

Abstract

The paper is concerned with a generalization of concepts introduced in [13], i.e. introduced are matrices of linear transformations over a finitedimensional vector space. Introduced are linear transformations over a finitedimensional vector space depending on a given matrix of the transformation. Finally, I prove that the rank of linear transformations over a finite-dimensional vector space is the same as the rank of the matrix of that transformation.

References

[1] Jesse Alama. The rank+nullity theorem. Formalized Mathematics, 15(3):137-142, 2007.10.2478/v10037-007-0015-6
Search in Google Scholar
[2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
Search in Google Scholar
[3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
Search in Google Scholar
[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
Search in Google Scholar
[5] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990.
Search in Google Scholar
[6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
Search in Google Scholar
[7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
Search in Google Scholar
[8] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
Search in Google Scholar
[9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
Search in Google Scholar
[10] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475-480, 1991.
Search in Google Scholar
[11] Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992.
Search in Google Scholar
[12] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
Search in Google Scholar
[13] Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339-345, 1996.
Search in Google Scholar
[14] Michał Muzalewski. Rings and modules - part II. Formalized Mathematics, 2(4):579-585, 1991.
Search in Google Scholar
[15] Karol Pαk. Basic properties of the rank of matrices over a field. Formalized Mathematics, 15(4):199-211, 2007.10.2478/v10037-007-0024-5
Search in Google Scholar
[16] Karol Pαk. Block diagonal matrices. Formalized Mathematics, 16(3):259-267, 2008.10.2478/v10037-008-0031-1
Search in Google Scholar
[17] Karol Pαk. Solutions of linear equations. Formalized Mathematics, 16(1):81-90, 2008.10.2478/v10037-008-0012-4
Search in Google Scholar
[18] Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883-885, 1990.
Search in Google Scholar
[19] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.
Search in Google Scholar
[20] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1(5):877-882, 1990.
Search in Google Scholar
[21] Wojciech A. Trybulec. Operations on subspaces in vector space. Formalized Mathematics, 1(5):871-876, 1990.
Search in Google Scholar
[22] Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865-870, 1990.
Search in Google Scholar
[23] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
Search in Google Scholar
[24] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
Search in Google Scholar
[25] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
Search in Google Scholar
[26] Xiaopeng Yue, Xiquan Liang, and Zhongpin Sun. Some properties of some special matrices. Formalized Mathematics, 13(4):541-547, 2005.
Search in Google Scholar
[27] Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205-211, 1992.
Search in Google Scholar
[28] Katarzyna Zawadzka. The product and the determinant of matrices with entries in a field. Formalized Mathematics, 4(1):1-8, 1993.
Search in Google Scholar
[29] Mariusz Żynel. The Steinitz theorem and the dimension of a vector space. Formalized Mathematics, 5(3):423-428, 1996.
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/v10037-008-0032-0 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630

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Language: English

Page range: 269 - 275

Published on: Mar 20, 2009

Published by: University of Białystok

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Related subjects:

Logic and set theory,

Computer sciences,

Artificial intelligence

© 2009 Karol Pąk, published by University of Białystok
This work is licensed under the Creative Commons License.

Volume 16 (2008): Issue 3 (September 2008)