Finding a base for a vector space of polynomials

In (Carre, 1979), and in other presentations the author proposed the definition of a fuzzy graph as a pair of vector spaces. The underlying reason for the departure from its usual definition as depicting a fuzzy relation, is not only the elegance of the new definition but also because of the practicality of having the tools of linear algebra available should we be able to find a basis for each vector space. If fuzziness is portrayed in the form of numbers then to find a base, the work of (Anton, 1987) is helpful. What if we wish to use functions such as polynomials? It is possible to show that under suitable conditions, a polynomial can serve as a fuzzy number. In (McAllister, 1992) we find help to solve the problem. This work illustrates how the task can be done. Why is this research important? Its importance relies on the fact that vector spaces are spanned by their bases. If these bases are known, then any vertex not functioning properly, or any link that is defective or inactive can be made fully functional by expressing them as a linear combination of the elements of its basis.