Fundamental set of solutions determines an unique homogeneous linear equation with constant coefficients

Question

Given $$ X_{1}(t) = (\exp(t),0,\exp(t)) ;X_{2}(t)=(2,1,0); X_{3}(t)=(0,2\exp(-t),-\exp(-t))$$ if I'm interested in a LHCC system that has these three functions as its solutions, then there would be only one system satisfying these conditions as there is only one linear transformation with the eigenvalues and associated eigenvectors as those shown above, so the associated system would be the matrix with respect to the standard basis with eigenvalues $1,0,-1$ and the eigenvectors $(1,0,1); (2,1,0), (0,2,-1)$ respectively. Is this approach correct?