# Basic Properties of the Rank of Matrices over a Field

Formalized Mathematics (2007)

• Volume: 15, Issue: 4, page 199-211
• ISSN: 1426-2630

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## Abstract

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In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field.I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entries constrained by row and column. Then I introduce the concept of the rank of a m x n matrix A by the condition: A has the rank r if and only if, there is a r x r submatrix of A with a non-zero determinant, and for every k x k submatrix of A with a non-zero determinant we have k ≤ r.At the end, I prove that the rank defined by the size of the biggest submatrix with a non-zero determinant of a matrix A, is the same as the maximal number of linearly independent rows of A.

## How to cite

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Karol Pąk. "Basic Properties of the Rank of Matrices over a Field." Formalized Mathematics 15.4 (2007): 199-211. <http://eudml.org/doc/267151>.

@article{KarolPąk2007,
abstract = {In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field.I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entries constrained by row and column. Then I introduce the concept of the rank of a m x n matrix A by the condition: A has the rank r if and only if, there is a r x r submatrix of A with a non-zero determinant, and for every k x k submatrix of A with a non-zero determinant we have k ≤ r.At the end, I prove that the rank defined by the size of the biggest submatrix with a non-zero determinant of a matrix A, is the same as the maximal number of linearly independent rows of A.},
author = {Karol Pąk},
journal = {Formalized Mathematics},
language = {eng},
number = {4},
pages = {199-211},
title = {Basic Properties of the Rank of Matrices over a Field},
url = {http://eudml.org/doc/267151},
volume = {15},
year = {2007},
}

TY - JOUR
AU - Karol Pąk
TI - Basic Properties of the Rank of Matrices over a Field
JO - Formalized Mathematics
PY - 2007
VL - 15
IS - 4
SP - 199
EP - 211
AB - In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field.I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entries constrained by row and column. Then I introduce the concept of the rank of a m x n matrix A by the condition: A has the rank r if and only if, there is a r x r submatrix of A with a non-zero determinant, and for every k x k submatrix of A with a non-zero determinant we have k ≤ r.At the end, I prove that the rank defined by the size of the biggest submatrix with a non-zero determinant of a matrix A, is the same as the maximal number of linearly independent rows of A.
LA - eng
UR - http://eudml.org/doc/267151
ER -

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