| The old and famous Lvov-Kaplansky conjecture asserts:The images of a multilinear polynomial in non-commutative variables over a field K on the matrix algebra Mn(K)is always a vector space.Over the years,many algebra scholars have studied this conjecture and obtained many results,but so far,it has not been completely solved,and it is not clear whether The images of multilinear polynomials in 2-order full matrix algebra in the general field is a linear space or not.The images of multilinear polynomials on 2 × 2 upper triangular matrix algebra over field K has been described.In this paper,we described The images of multilinear polynomials on 3 × 3 upper triangular matrix algebra over field K.That is,let UT3 is a 3×3 upper triangular matrix algebra over field K and f(x1,...,xm)is a multilinear polynomial in non-conmmutative variables over K,if |K|>3,then f(UT3)is a vector space over K. |
