Space of polynomials of bounded degree

Example

Let $F$ be a field and let $n \ge 0$ be an integer. Let $F[x]_{<n}$ be the set of polynomials in $x$ with coefficients in $F$ and of degree less than $n$, with the zero polynomial included. It is a vector space over $F$, with the usual addition of polynomials and multiplication by constants. The monomials $1, x, x^2, \dots, x^{n-1}$ form a basis, so $\dim F[x]_{<n} = n$.

Remarks

The set of polynomials of degree exactly $n$ is not a vector space. It does not contain $0$, and it is not closed under addition. The whole polynomial ring $F[x]$ is an infinite-dimensional vector space with basis $1, x, x^2, \dots$. Differentiation is a linear map from $F[x]_{<n}$ to $F[x]_{<n-1}$ for $n \ge 1$.

Explanations

L1 human

Polynomials of small degree can be added and scaled, and the result stays a polynomial of small degree. So they form a vector space. A polynomial $a_0 + a_1 x + a_2 x^2$ is the same data as the list $(a_0, a_1, a_2)$.