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 \(\mathbb{R}^3\).Copy