--- Commentary: $\textbf{Exercise 3.}$ (a) Let $U = {p \in \mathcal{P}_4(\mathbb{F}) : p(6) = 0}$. Find a basis of $U$. (b) Extend the basis in (a) to a basis of $\mathcal{P}_4(\mathbb{F})$. (c) Find a subspace $W$ of $\mathcal{P}_4(\mathbb{F})$ such that $\mathcal{P}_4(\mathbb{F}) = U \oplus W$. $\textbf{Solution 13.}$ (a) A basis of $U$ is $x - 6, (x - 6)^2, (x - 6)^3, (x - 6)^4.$ To verify that the list above is indeed a basis of $U$, first note that the list above is linearly independent (using the same reasoning as was used in Example 2.41 to show that the list in that example is linearly independent). Then note that the linearly independent list above has length four, and thus $\dim U \geq 4$. However, $\dim \mathcal{P}_4(\mathbb{F}) = 5$, which implies that $\dim U = 4$ or $\dim U = 5$. Because $U$ is a proper subspace of $\mathcal{P}_4(\mathbb{F})$, this implies that $\dim U = 4$. Hence the list above is a basis of $U$. (b) The constant function 1 clearly is not in $U$. Thus $x - 6, (x - 6)^2, (x - 6)^3, (x - 6)^4, 1$ is a linearly independent list in $\mathcal{P}_4(\mathbb{F})$ of length five. By 2.38, the list above is a basis of $\mathcal{P}_4(\mathbb{F})$. (c) Using the idea of the proof of 2.33 and the answer above to (b), we see that taking $W$ to be the subspace of $\mathcal{P}_4(\mathbb{F})$ consisting of the constant functions gives a subspace $W$ such that $\mathcal{P}_4(\mathbb{F}) = U \oplus W$. $\textit{Commentary:}$ This exercise deals with a subspace $U$ of $\mathcal{P}_4(\mathbb{F})$, the space of polynomials of degree at most 4 over a field $\mathbb{F}$. The subspace $U$ consists of all polynomials in $\mathcal{P}_4(\mathbb{F})$ that have a zero at $x = 6$. In part (a), a basis for $U$ is found. The polynomials $(x - 6)^k$ for $k = 1, 2, 3, 4$ are chosen. These polynomials clearly lie in $U$ as they all have a zero at $x = 6$. They are linearly independent for the same reason that $1, x, x^2, x^3, x^4$ are linearly independent: a polynomial of degree $n$ cannot be a linear combination of polynomials of lower degree. Finally, the dimension of $U$ is argued to be 4 by considering the dimension of the whole space $\mathcal{P}_4(\mathbb{F})$, which is 5. Since $U$ is a proper subspace, its dimension must be less than 5, and since we have found 4 linearly independent vectors in $U$, its dimension must be at least 4, hence equal to 4. In part (b), the basis of $U$ is extended to a basis of $\mathcal{P}_4(\mathbb{F})$ by adding the constant polynomial 1, which is not in $U$. In part (c), the subspace $W$ is chosen to be the span of the polynomial added in part (b), which is the space of constant polynomials. This ensures that $U$ and $W$ intersect only at the zero polynomial and that every polynomial in $\mathcal{P}_4(\mathbb{F})$ can be uniquely written as a sum of a polynomial in $U$ and a constant polynomial. This exercise illustrates the process of finding a basis for a subspace defined by a condition (here, having a zero at a specific point), extending this basis to a basis of the whole space, and identifying a complementary subspace for a direct sum decomposition. $\textit{Examples:}$ 1. In $\mathcal{P}_5(\mathbb{R})$, let $U = {p \in \mathcal{P}_5(\mathbb{R}) : p(1) = p'(1) = 0}$. A basis for $U$ is $(x - 1)^2, (x - 1)^3, (x - 1)^4, (x - 1)^5$. This can be extended to a basis of $\mathcal{P}_5(\mathbb{R})$ by adding $1$ and $x - 1$. The subspace $W = \operatorname{span}(1, x - 1)$ satisfies $\mathcal{P}_5(\mathbb{R}) = U \oplus W$. 2. In $\mathcal{P}_3(\mathbb{C})$, let $U = {p \in \mathcal{P}_3(\mathbb{C}) : p(i) = 0}$. A basis for $U$ is $x - i, (x - i)^2, (x - i)^3$. This can be extended to a basis of $\mathcal{P}_3(\mathbb{C})$ by adding 1. The subspace $W = \operatorname{span}(1)$ satisfies $\mathcal{P}_3(\mathbb{C}) = U \oplus W$. 3. In $\mathcal{P}_4(\mathbb{F}_2)$, let $U = {p \in \mathcal{P}_4(\mathbb{F}_2) : p(0) = p(1) = 0}$. A basis for $U$ is $x(x - 1), x(x - 1)^2, x(x - 1)^3$. This can be extended to a basis of $\mathcal{P}_4(\mathbb{F}_2)$ by adding 1 and $x$. The subspace $W = \operatorname{span}(1, x)$ satisfies $\mathcal{P}_4(\mathbb{F}_2) = U \oplus W$. These examples demonstrate similar principles in polynomial spaces over different fields, with subspaces defined by different conditions. The conditions can involve the values of the polynomials or their derivatives at certain points. The key steps remain the same: find a basis for the subspace, extend it to a basis of the whole space, and identify a complementary subspace. ---