--- Commentary: $\textbf{Exercise 14.}$ Suppose $U = {(x, x, y, y) \in \mathbb{F}^4 : x, y \in \mathbb{F}}.$ Find a subspace $W$ of $\mathbb{F}^4$ such that $\mathbb{F}^4 = U \oplus W$. $\textbf{Solution 14.}$ Let $W = {(a, 0, b, 0) \in \mathbb{F}^4 : a, b \in \mathbb{F}}.$ Then $\mathbb{F}^4 = U \oplus W$, as is easy to verify. $\textit{Commentary:}$ This exercise asks to find a subspace $W$ such that $\mathbb{F}^4$ is the direct sum of $U$ and $W$. The given subspace $U$ consists of vectors whose first and second coordinates are equal, and whose third and fourth coordinates are equal. To find $W$, we look for a subspace that is "complementary" to $U$ in the sense that every vector in $\mathbb{F}^4$ can be uniquely written as a sum of a vector in $U$ and a vector in $W$. One way to do this is to choose $W$ to be the subspace of vectors whose second and fourth coordinates are 0. This ensures that the only vector common to $U$ and $W$ is the zero vector, and that any vector in $\mathbb{F}^4$ can be written as a sum of a vector in $U$ and a vector in $W$. This exercise illustrates the concept of a direct sum decomposition of a vector space, which is a way of breaking a vector space into simpler pieces (subspaces) that are "independent" from each other. $\textit{Example:}$ In $\mathbb{Q}_p^4$, let $U = {(x, px, y, py) : x, y \in \mathbb{Q}_p}$. Then $W = {(a, 0, b, 0) : a, b \in \mathbb{Q}_p}$ satisfies $\mathbb{Q}_p^4 = U \oplus W$. In $\mathbb{F}_3^4$, let $U = {(x, x, y, y) : x, y \in \mathbb{F}_3}$. Then $W = {(a, 2a, b, 2b) : a, b \in \mathbb{F}_3}$ satisfies $\mathbb{F}_3^4 = U \oplus W$. Note that in $\mathbb{F}_3$, $2 = -1$, so the condition defining $W$ can be written as $(a, -a, b, -b)$. In the space of polynomials $\mathbb{R}[x]$, let $U$ be the subspace of polynomials where the coefficients of odd degree terms are equal to the coefficients of the next even degree terms. For example, $1 + 2x + 2x^2 + 3x^3 + 3x^4 \in U$. Then $W$, the subspace of polynomials where the coefficients of odd degree terms are the negatives of the coefficients of the next even degree terms, satisfies $\mathbb{R}[x] = U \oplus W$. These examples demonstrate the process of finding a direct sum complement in various vector spaces over different fields. The key is to choose $W$ in a way that ensures its intersection with $U$ is ${0}$ and that every vector can be uniquely decomposed into a sum of vectors from $U$ and $W$. The specific choice of $W$ depends on the structure of $U$ and the properties of the underlying field.