  The Fagundes-Mello conjecture asserts that every multilinear polynomial on
upper triangular matrix algebras is a vector space, which is an improtant
variation of the old and famous Lvov-Kaplansky conjecture. The goal of the
paper is to give a description of the images of linear polynomials with zero
constant term on the upper triangular matrix algebra under a mild condition on
the ground field. As a consequence we improve all results on the Fagundes-Mello
conjecture. As another consequence we improve some results by Fagundes and
Koshlukov on the images of multilinear graded polynomials on upper triangular
matrix algebras.
