This needs an expansion: Direct sum definition explanation how adding two more dimensions, a, b, to x, y fills out $F^4$ see 21  --- Commentary: $\textbf{Exercise 20.}$ Suppose $U = {(x_1, x_2, x_3, x_4) \in \mathbb{F}^4 : x_1 = x_2 \text{ and } x_3 = x_4}.$ Find a subspace $W$ of $\mathbb{F}^4$ such that $\mathbb{F}^4 = U \oplus W$. $\textbf{Solution 20.}$ Let $W = {(a, -a, b, -b) \in \mathbb{F}^4 : a, b \in \mathbb{F}}.$ Then $\mathbb{F}^4 = U \oplus W$, as is easy to verify. $\textit{Commentary:}$ This exercise is similar to Exercise 14, asking 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. The solution chooses $W$ to be the subspace of vectors whose second coordinate is the negative of the first, and whose fourth coordinate is the negative of the third. This choice ensures that $U \cap W = {0}$, because the only vector that satisfies both the conditions for $U$ and the conditions for $W$ is the zero vector. Moreover, any vector in $\mathbb{F}^4$ can be uniquely written as a sum of a vector in $U$ and a vector in $W$, by taking the average of the first two coordinates and the average of the last two coordinates for the $U$ component, and half the difference of the first two coordinates and half the difference of the last two coordinates for the $W$ component. This exercise reinforces the concept of a direct sum decomposition and provides practice in finding a complementary subspace to a given subspace. $\textit{Additional Examples:}$ In $\mathbb{R}^4$, if $U = {(x, x, y, y) : x, y \in \mathbb{R}}$, then $W = {(a, -a, b, -b) : a, b \in \mathbb{R}}$ satisfies $\mathbb{R}^4 = U \oplus W$. In $\mathbb{C}^4$, if $U = {(z, z, w, w) : z, w \in \mathbb{C}}$, then $W = {(a, -a, b, -b) : a, b \in \mathbb{C}}$ satisfies $\mathbb{C}^4 = U \oplus W$. In $\mathbb{F}_2^4$, if $U = {(x, x, y, y) : x, y \in \mathbb{F}_2}$, then $W = {(a, a+1, b, b+1) : a, b \in \mathbb{F}_2}$ satisfies $\mathbb{F}_2^4 = U \oplus W$. Note that in $\mathbb{F}_2$, $-1 = 1$. In the space of functions from $\mathbb{R}$ to $\mathbb{R}$, if $U$ is the subspace of functions $f$ satisfying $f(x) = f(-x)$ for all $x$ (even functions), then $W$, the subspace of functions $f$ satisfying $f(x) = -f(-x)$ for all $x$ (odd functions), satisfies the space of functions = $U \oplus W$. These additional examples illustrate the same principle in different contexts, including over the complex numbers, over finite fields, and in a function space. The key idea remains to choose $W$ in a way that ensures it intersects $U$ only at the zero vector and that every vector can be uniquely decomposed into a sum from $U$ and $W$.