I am using "Linear Algebra Done Right" as a self study guide, and was confused by the following question in the text:
For $m$ a nonnegative integer, let $P_m(F)$ denote the set of all polynomials with coefficients in $F$ and degree at most $m$. You should verify that $P_m(F)$ is a subspace of $P(F)$ [where $P(F)$ is the space of all polynomials with coefficients in $F$]
The book's defines degree as:
A polynomial $p \in P(F)$ is said to have degree $m$ if there exists scalars $a_0, a_1,...a_m$ with $a_m \neq 0$ such that $p(z)=a_0+a_1z+...+a_mz^m$
For $P_m$ to be a subset of $P(F)$, $P_m$ must contain the additive inverse of $P(F)$, which is the polynomial with all 0 coefficients (according to the book). If my understanding of all this is correct, then isn't $P_0$, and subsequently any $P_m$, not a subspace of $P(F)$ because the definition of polynomials of degree $m$ requires at least one non-zero coefficient?