Exercise 2.C.8

Question

(a) Let $U = \{p \in \mathcal{P}_4 (R): \int_{-1}^1 p = 0\}$. Find a basis of U.

(b) Extend the basis in part (a) to a basis of $\mathcal{P}_4 (R)$.

(c) Find a subspace $W$ of $\mathcal{P}_4 (R)$ such that $\mathcal{P}_4 (R) = U \bigoplus W.

Arden Shi · Accepted Answer

(a) given the condition that $\int_{-1}^1 p = 0$, we can conclude that $p$ must be an odd function. For a polynomial to be odd, it must only have odd powers of $x$ in it. Therefore, $p = a_1x^3+a_2x$. Thus, $x$, and $x^3$ are a basis of $U$.

(b) To extend it, we only have to add on $1$,  $x^2$, and $x^4$ to make it be a basis of $\mathcal{P}_4 (R)$.

(c) 
$$
W = \{p \in \mathcal{P}_4 (R): \int_{-1}^0 p = \int_0^1 p\}
$$

Exercise 2.C.8

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Exercise 2.C.8

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