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.
@article{bwmeta1.element.doi-10_2478_v10037-007-0024-5,
author = {Karol P\k ak},
title = {Basic Properties of the Rank of Matrices over a Field},
journal = {Formalized Mathematics},
volume = {15},
year = {2007},
pages = {199-211},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10037-007-0024-5}
}
Karol Pąk. Basic Properties of the Rank of Matrices over a Field. Formalized Mathematics, Tome 15 (2007) pp. 199-211. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10037-007-0024-5/
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