Linear Map of Matrices

Karol Pąk

Formalized Mathematics (2008)

  • Volume: 16, Issue: 3, page 269-275
  • ISSN: 1426-2630

Abstract

top
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.

How to cite

top

Karol Pąk. "Linear Map of Matrices." Formalized Mathematics 16.3 (2008): 269-275. <http://eudml.org/doc/267194>.

@article{KarolPąk2008,
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.},
author = {Karol Pąk},
journal = {Formalized Mathematics},
language = {eng},
number = {3},
pages = {269-275},
title = {Linear Map of Matrices},
url = {http://eudml.org/doc/267194},
volume = {16},
year = {2008},
}

TY - JOUR
AU - Karol Pąk
TI - Linear Map of Matrices
JO - Formalized Mathematics
PY - 2008
VL - 16
IS - 3
SP - 269
EP - 275
AB - 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.
LA - eng
UR - http://eudml.org/doc/267194
ER -

References

top
  1. [1] Jesse Alama. The rank+nullity theorem. Formalized Mathematics, 15(3):137-142, 2007. 
  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 and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990. 
  5. [5] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990. 
  6. [6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990. 
  7. [7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990. 
  8. [8] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990. 
  9. [9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990. 
  10. [10] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475-480, 1991. 
  11. [11] Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992. 
  12. [12] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990. 
  13. [13] Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339-345, 1996. 
  14. [14] Michał Muzalewski. Rings and modules - part II. Formalized Mathematics, 2(4):579-585, 1991. 
  15. [15] Karol Pαk. Basic properties of the rank of matrices over a field. Formalized Mathematics, 15(4):199-211, 2007. 
  16. [16] Karol Pαk. Block diagonal matrices. Formalized Mathematics, 16(3):259-267, 2008. 
  17. [17] Karol Pαk. Solutions of linear equations. Formalized Mathematics, 16(1):81-90, 2008. 
  18. [18] Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883-885, 1990. 
  19. [19] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990. 
  20. [20] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1(5):877-882, 1990. 
  21. [21] Wojciech A. Trybulec. Operations on subspaces in vector space. Formalized Mathematics, 1(5):871-876, 1990. 
  22. [22] Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865-870, 1990. 
  23. [23] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990. 
  24. [24] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. 
  25. [25] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990. 
  26. [26] Xiaopeng Yue, Xiquan Liang, and Zhongpin Sun. Some properties of some special matrices. Formalized Mathematics, 13(4):541-547, 2005. 
  27. [27] Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205-211, 1992. 
  28. [28] Katarzyna Zawadzka. The product and the determinant of matrices with entries in a field. Formalized Mathematics, 4(1):1-8, 1993. 
  29. [29] Mariusz Żynel. The Steinitz theorem and the dimension of a vector space. Formalized Mathematics, 5(3):423-428, 1996. 

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.