Relationships Among The Different Definitions Of Linear Independence And Invertible Matrices Over Commutative Semirings

Abstrct In1996, Cechldrova and Plavka suggested several possible definitions of linear independence of sets of n-tuples in bottleneck algebras and studied their characterizations and interrelations. This paper first introduces several possible definitions of linear independence of sets of n-tuples over commutative semirings, discusses their interrelations, then proves that the set of the column vectors of a square matrix is linear independence if the matrix is invertible. It also obtains the relationship between an invertible matrix and the linear independence of set of row (resp. column) vectors of the matrix over commutative zerosumfree semirings. Finally, it shows that an n×n matrix is invertible if and only if the rank of the matrix is n.