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