--- Commentary: $\textbf{Exercise 21.}$ Suppose $U = {(x, y, x+y, x-y, 2x) \in \mathbb{F}^5 : x, y \in \mathbb{F}}.$ Find a subspace $W$ of $\mathbb{F}^5$ such that $\mathbb{F}^5 = U \oplus W$. $\textbf{Solution 21.}$ Let $W = {(0, 0, a, b, c) \in \mathbb{F}^5 : a, b, c \in \mathbb{F}}.$ Then $\mathbb{F}^5 = U \oplus W$, as is easy to verify. $\textit{Commentary:}$ This exercise follows the same pattern as Exercises 14 and 20, but in the context of $\mathbb{F}^5$. The given subspace $U$ consists of vectors where the third coordinate is the sum of the first two, the fourth coordinate is the difference of the first two, and the fifth coordinate is twice the first. The solution chooses $W$ to be the subspace of vectors whose first two coordinates are 0. This ensures that $U \cap W = {0}$, because the only vector in $U$ with first two coordinates 0 is the zero vector. Moreover, any vector in $\mathbb{F}^5$ can be uniquely written as a sum of a vector in $U$ (by choosing the first two coordinates and computing the rest) and a vector in $W$ (by taking the last three coordinates). This exercise provides further practice in finding a direct sum complement and illustrates how the method extends to higher-dimensional spaces. $\textit{Additional Examples:}$ In $\mathbb{Q}^5$, if $U = {(x, y, x+y, x-y, 2x) : x, y \in \mathbb{Q}}$, then $W = {(0, 0, a, b, c) : a, b, c \in \mathbb{Q}}$ satisfies $\mathbb{Q}^5 = U \oplus W$. In $\mathbb{F}_3^5$, if $U = {(x, y, x+y, x-y, 2x) : x, y \in \mathbb{F}_3}$, then $W = {(0, 0, a, b, c) : a, b, c \in \mathbb{F}_3}$ satisfies $\mathbb{F}_3^5 = U \oplus W$. Note that in $\mathbb{F}_3$, $-1 = 2$, so $x-y$ really means $x+2y$. In the space of polynomials $\mathbb{R}[x,y]$, if $U = {ax+by+c(x+y)+d(x-y)+e(2x) : a,b,c,d,e \in \mathbb{R}}$, then $W = {f+gy+hz : f,g,h \in \mathbb{R}[x,y]}$ satisfies $\mathbb{R}[x,y] = U \oplus W$. In the space of matrices $M_{2\times 3}(\mathbb{C})$, if $U = \left{\begin{pmatrix}a & b & a+b \ c & d & c-d\end{pmatrix} : a,b,c,d \in \mathbb{C}\right}$, then $W = \left{\begin{pmatrix}0 & 0 & e \ 0 & 0 & f\end{pmatrix} : e,f \in \mathbb{C}\right}$ satisfies $M_{2\times 3}(\mathbb{C}) = U \oplus W$. These examples demonstrate the same principle in various vector spaces, including - over the rational numbers, - over finite fields, - in a polynomial ring, and - in a matrix space. - - The specific conditions defining $U$ and $W$ may vary depending on the space and field, but the underlying idea of choosing $W$ to be a "complementary" subspace to $U$ remains the same.