Linear Transformations of Euclidean Topological Spaces

Karol Pąk

Formalized Mathematics (2011)

  • Volume: 19, Issue: 2, page 103-108
  • ISSN: 1426-2630

Abstract

top
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.

How to cite

top

Karol Pąk. "Linear Transformations of Euclidean Topological Spaces." Formalized Mathematics 19.2 (2011): 103-108. <http://eudml.org/doc/266875>.

@article{KarolPąk2011,
abstract = {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.},
author = {Karol Pąk},
journal = {Formalized Mathematics},
language = {eng},
number = {2},
pages = {103-108},
title = {Linear Transformations of Euclidean Topological Spaces},
url = {http://eudml.org/doc/266875},
volume = {19},
year = {2011},
}

TY - JOUR
AU - Karol Pąk
TI - Linear Transformations of Euclidean Topological Spaces
JO - Formalized Mathematics
PY - 2011
VL - 19
IS - 2
SP - 103
EP - 108
AB - 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.
LA - eng
UR - http://eudml.org/doc/266875
ER -

References

top
  1. [1] Jesse Alama. The rank+nullity theorem. Formalized Mathematics, 15(3):137-142, 2007, doi:10.2478/v10037-007-0015-6.[Crossref] 
  2. [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990. 
  3. [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
  4. [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. 
  5. [5] Grzegorz Bancerek, Mitsuru Aoki, Akio Matsumoto, and Yasunari Shidama. Processes in Petri nets. Formalized Mathematics, 11(1):125-132, 2003. 
  6. [6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990. 
  7. [7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990. 
  8. [8] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990. 
  9. [9] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990. 
  10. [10] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990. 
  11. [11] Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990. 
  12. [12] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991. 
  13. [13] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23-28, 2003. 
  14. [14] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990. 
  15. [15] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475-480, 1991. 
  16. [16] Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181-187, 2005. 
  17. [17] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990. 
  18. [18] Anna Lango and Grzegorz Bancerek. Product of families of groups and vector spaces. Formalized Mathematics, 3(2):235-240, 1992. 
  19. [19] Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339-345, 1996. 
  20. [20] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990. 
  21. [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.[Crossref] 
  22. [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.[Crossref] 
  23. [23] Karol Pąk. Solutions of linear equations. Formalized Mathematics, 16(1):81-90, 2008, doi:10.2478/v10037-008-0012-4.[Crossref] 
  24. [24] Karol Pąk. Linear map of matrices. Formalized Mathematics, 16(3):269-275, 2008, doi:10.2478/v10037-008-0032-0.[Crossref] 
  25. [25] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990. 
  26. [26] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990. 
  27. [27] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. 
  28. [28] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990. 
  29. [29] Xiaopeng Yue, Xiquan Liang, and Zhongpin Sun. Some properties of some special matrices. Formalized Mathematics, 13(4):541-547, 2005. 
  30. [30] Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205-211, 1992. 
  31. [31] Katarzyna Zawadzka. The product and the determinant of matrices with entries in a field. Formalized Mathematics, 4(1):1-8, 1993. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.