Question 4.1.16: Find a basis of P2 , the space of all polynomials of degree ......

We can write any polynomial f (x) of degree ≤2 as

f(x) = a + bx + cx^2 = a  · 1 + b  · x + c  · x^2,

showing that the monomials 1, x, x^2 span P_2. We leave it as an exercise for the reader to verify the linear independence of these monomials. Thus, 𝔅= (1, x, x^2) is a basis (called the standard basis of P_2), so that dim(P_2) = 3.

The 𝔅-coordinate transformation L_𝔅 is represented in the following diagram:

f (x) = a + bx + cx^2\; in P_2\xrightarrow{L_𝔅}[f(x)]_𝔅=\begin{bmatrix}a\\b\\c\end{bmatrix}\; in ℝ^3.