As another demonstration of using Sage to help us understand spans, linear combinations, linear independence and reduced row-echelon form, we will recreate parts of Example RES. Most of this should be familiar, but see the commments following.
The final two results — a trivial null space for coeff and the linear independence of P — both individually imply that the solution to the system of equations (just prior) is unique. Sage produces its own linearly independent spanning set for each span, as we see whenever we inquire about a span.
Can you extract the three vectors that Sage uses to span X and solve the appropriate system of equations to see how to write y as a linear combination of these three vectors? Once you have done that, check your answer by hand and think about how using Sage could have been overkill for this question.