  A solution of linear systems of equations Ax=b and Ax=0 is a vital part of
many computational packages. This paper presents a novel formulation based on
the projective extension of the Euclidean space using the outer product
(extended cross-product). This approach enables to solve the both cases, i.e.
Ax=b and Ax=0. The proposed approach leads actually to an analytical solution
of linear systems in the form on which the other vector operation can be
applied before using the numerical evaluation.
  This contribution also proposes a new approach to the conditionality
estimation of matrices applicable to non-squared matrices. It splits the
conditionality to structural conditionality showing matrix property if nearly
unlimited precision is used, numerical issue which depends on numerical
representation with respect to the right-hand side influence, if given.
