Some Problems Of Linear Algebras Over Zerosumfree Semirings

This paper deals with the characterizations of a basis in semilinear space over zerosumfree semirings and their applications in the dimensions of semilinear spaces, the solution of systems of equations, the relationship among the semilinear subspaces and the bideterminant. Firstly, in L-semilinear space Vn, some necessary and sufficient conditions that each basis has the same number of elements and a set of linearly independent vectors is a basis are given. It is shown that a set of linearly independent vectors can be extended to a basis. In the following, some characterizations of standard orthogonal vectors are investigated. As applications, some necessary and sufficient conditions that a set of vectors is a basis of an L-semilinear subspace which is generated by standard orthogonal vectors, a set of linearly independent non standard orthogonal vectors can not be orthogonalized if it has at least two nonzero vectors and the analog of the Kronecker-Capelli theorem is valid for systems of equations are formulated in L-semilinear space Vn. Concepts of semi-linearly dependent and strong linearly independent of a set of vectors are defined. Some necessary and sufficient conditions that the sum of L-semilinear subspace of Vn is direct sum each basis has the same number of elements are shown. In particular, from the algebraic structure point of view, a necessary and sufficient condition that two semilinear spaces are isomorphic is presented. In the end, some characterizations of bidetermiant are investigated. Over some spacial zerosumfree semirings, it is proven that the bidetermiant of a matrix is equal to0if and only if the column-vectors of this matrix are linearly dependent or semi-linearly dependent. Furthermore, dimension of semilinear subspace can be defined over some spacial zerosumfree semirings and a necessary and sufficient condition that the rank of n-square matrix is equal to n are obtained.