Define linear combinations of matrix columns with fuzzy coefficients in the Gaussian-PDMF framework

Develop a rigorous and operational definition of linear combinations of the columns of a real matrix using coefficients from the Gaussian Probability Density Membership Function (Gaussian-PDMF) space X of fuzzy numbers, so that column-space style constructions can be meaningfully formulated within the fuzzy linear algebra framework used for semi-fuzzy and fully-fuzzy linear systems.

Background

In the classical setting, the null space can be characterized via orthogonality to the column space, which is defined through linear combinations of columns with scalar coefficients. In the Gaussian-PDMF framework, the authors consider solving semi-fuzzy and fully-fuzzy linear systems where coefficients and/or unknowns are fuzzy numbers from the commutative ring X.

However, extending column-space concepts requires defining linear combinations of matrix columns with fuzzy coefficients from X. The authors explicitly state that this notion remains unresolved in their paper, preventing the direct transfer of certain classical linear-algebraic characterizations (such as the orthogonal complement relation) to the fuzzy setting.