# 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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topKarol 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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