No, it is not possible to create $\vect{w}$ as a linear combination
of the four vectors $\vect{u}_1,\,\vect{u}_2,\,\vect{u}_3,\,\vect{u}_4$. By creating the desired linear combination with unknowns as scalars, Theorem SLSLC provides a system of equations that has no solution. This one computation is enough to show us that it is not possible to create all the vectors of $\complex{6}$ through linear combinations of the four vectors $\vect{u}_1,\,\vect{u}_2,\,\vect{u}_3,\,\vect{u}_4$.