Does the typical vector space of polynomials up to a certain degree need to have the constant vectors

Question

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

Answer 1

If I understand you correctly, you are asking two separate questions:

1. Does the set of polynomials of degree $\le n$ have to include constant polynomials?
2. Is the set of polynomials of degree $\le n$ with constant term $=0$ a vector space?

The answer to the first question is Yes; by definition "polynomials of degree $\le n$" includes polynomials of degree $0$, which means that the set of polynmials of degree $\le n$ must include constants.

However, I think you know that already, and I think what you really want to know is the answer to the second question, which is also Yes. One can certainly consider the set of polynomials of degree $\le n$ whose constant terms are $0$, and verify (as you seem to have done) that it satisfies all of the requirements for a vector space. This subspace of $P_n$ can also be described as the subset of $P_n$ consisting of polynomials for which $x$ is a factor.

To bring this back to the question that started the whole thing: the exercise that prompted this asked "Does this set of polynomials span $P_4$?" The answer to that question is no, because $P_4$ contains constant polynomials, and there is no way to get a constant polynomial as a linear combination of the given set of polynomials. It is true that the set of polynomials does span a vector space -- but that vector space is not all of $P_4$. In fact, you can be sure it is not, because (as you have already determined) the vector space spanned by those vectors is 4-dimensional, while $P_4$ is 5-dimensional.