Standard orthogonal vectors in semilinear spaces and their applications (Q1758474)

An algebraic structure \(L\) with two binary operations \(+,\cdot\) and containing \(0\neq1\) is a commutative semiring if \((L,+,0)\) and \((L,\cdot,1)\) are commutative monoids, \(r\cdot0=0\cdot r=0\) for all \(r\in L\), and the distributive law holds. We shall also assume that the semiring is zerosumfree: \(r+s=0\implies r=s=0\). Consider the set \(V_{n}(L)\) of all (column) \(n\)-vectors and the set \(M_{m\times n}(L)\) of all \(m\times n\) matrices over \(L\). The authors investigate some of the elementary linear algebra in this generalized situation. Generators of a subspace and linear independence can be defined in a natural way and a basis of a subspace is a set of linearly independent generators. However, linear independence is a weak a notion in this context, and so (for example) bases may have different sizes. In [Linear Algebra Appl. 435, No. 11, 2681--2692 (2011; Zbl 1239.15002)], the authors introduced the notion of standard orthogonal vectors, namely, vectors which are mutually orthogonal under the usual dot product and whose dot product with themselves is a unit in \(L\). Here they show that (assuming \(1\) is an additive irreducible element of \(L\)): if \(W\) is a subspace of \(V_{n}(L)\) which is generated by a set of \(s\) standard orthogonal vectors, then every basis of \(W\) consists of \(s\) standard orthogonal vectors; in particular, we can define \(\dim W=s\). Furthermore, assume that every nonzero element of \(L\) is a unit, and that the columns \(a_{1},\dots,a_{m}\) of the square matrix \(A\) are standard orthogonal. Then the system \(Ax=b\) of linear equations is solvable over \(L\) if and only if the matrix \(A\) and the matrix \(A|b\) obtained by extending \(A\) by \(b\) have the same rank; in the latter case the solution is unique.