The set of all polynomials of degree at most n is a vector space.

Theorem: The set (P_n) of all polynomials of degree at most (n) with coefficients in (\mathbb{F}) is a vector space over (\mathbb{F}).

[P_n = {a_0 + a_1x + a_2x^2 + \cdots + a_nx^n : a_i \in \mathbb{F}}]

Basis: ({1, x, x^2, \ldots, x^n}) (the standard basis).

Dimension: (\dim(P_n) = n + 1).

Verification: The sum of two polynomials of degree at most (n) has degree at most (n), and a scalar multiple of such a polynomial also has degree at most (n).

Example: (P_2 = {a + bx + cx^2 : a,b,c \in \mathbb{R}}) has basis ({1, x, x^2}) and dimension 3. It is isomorphic to (\mathbb{R}^3).