I helped a classmate with an exercise about sets of vectors spanning (or not) certain vector spaces. The question was as follows: Does the set $$\{ t, t+t^2, t^3-3t^4, 2t-t^2, -4t^4-t^3 \}$$ span the space $$P_4$$ where this vector space was the polynomials with degree $\leq 4$. (where $t \in \mathbb{R}$)
(this vector space is defined the way it usually is, with coordinates of a vector corresponding with the coefficients of a polynomial)
I said yes because when you write the vectors as a $5$ by $4$ matrix you can use gaussian elimination to end up with 4 monomial vectors (by that I mean the vectors $(1,0,0,0)$ and so on which you would then use as the coefficients of a polynomial). This answer was wrong because the set does not include a constant vector.
Which brings to me to my actual question as to why specifically this is. As far as I have tried you could talk about a vector space of polynomials with degree $\leq k$ and no constant part just fine. I have tried finding where the vector space axioms don't work but I can't seem to find an error.
So is there something I am overlooking where a vector space of polynomials without a constant part would not work? I understand it would be pretty strange since if you're talking about polynomials there is no reason to disregard constant polynomials, especially since they don't cause any problems here, but let's disregard that for a moment.
P.S. yes I corrected my mistake to the classmate