The first thing you need to do is find a way of expressing this transformation. Express v1 given u1, v2 given u2 and v3 given u3. From there you can construct the matrix that you want, which will satisfy T(u)=Au=v.

Matrix A times the column vector [u1,u2,u3] will equal [v1,v2,v3,v4] (imagine those are columns). So A must be a 4x3 matrix and will be multiplied by the 3x1 column matrix for u, resulting in a 4x1 column matrix representing v. Each row in A will [a1,a2,a3] such that a1u1+a2u2+a3u3=v1.