Karol Pąk (2008)
Formalized Mathematics
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In this paper I present the Kronecker-Capelli theorem which states that a system of linear equations has a solution if and only if the rank of its coefficient matrix is equal to the rank of its augmented matrix.MML identifier: MATRIX15, version: 7.8.09 4.97.1001
Yatsuka Nakamura, Kunio Oniumi, Wenpai Chang (2008)
Formalized Mathematics
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In this paper the theory of invertibility of matrices of field elements (see e.g. [5], [6]) is developed. The main purpose of this article is to prove that the left invertibility and the right invertibility are equivalent for a matrix of field elements. To prove this, we introduced a special transformation of matrix to some canonical forms. Other concepts as zero vector and base vectors of field elements are also introduced as a preparation.MML identifier: MATRIX14, version: 7.9.01 4.101.1015 ...
Karol Pąk (2008)
Formalized Mathematics
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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.
Karol Pąk (2011)
Formalized Mathematics
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We prove a number of theorems concerning various notions used in the theory of continuity of barycentric coordinates.
Meshkov, Anatoly G., Balakhnev, Maxim Ju. (2008)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
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Dimakis, Aristophanes, Müller-Hoissen, Folkert (2009)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
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Krzysztof Moszyński (1995)
Applicationes Mathematicae
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Neuwirth, Erich (2002)
Séminaire Lotharingien de Combinatoire [electronic only]
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A. R. Moghaddamfar, S. M. H. Pooya, S. Navid Salehy, S. Nima Salehy (2012)
Matematički Vesnik
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Artur Korniłowicz (2010)
Formalized Mathematics
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We prove that basic arithmetic operations preserve continuity of functions.
Artur Korniłowicz (2010)
Formalized Mathematics
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In the article we prove that a family of open n-hypercubes is a basis of n-dimensional Euclidean space. The equality of the space and the product of n real lines has been proven.