--- Commentary: $\textbf{Exercise 19.}$ Prove or give a counterexample: If $V_1, V_2, U$ are subspaces of $V$ such that $V_1 + U = V_2 + U,$ then $V_1 = V_2$. $\textbf{Solution 19.}$ The statement above is false. To construct a counterexample, choose $V$ to be any nonzero vector space. Let $V_1 = {0}$, $V_2 = V$, and $U = V$. Then $V_1 + U$ and $V_2 + U$ are both equal to $V$, but $V_1 \neq V_2$. $\textit{Commentary:}$ This exercise asks whether the equation $V_1 + U = V_2 + U$ implies that $V_1 = V_2$ for subspaces $V_1, V_2, U$ of a vector space $V$. The answer is no, and the counterexample provided demonstrates why. The counterexample chooses $V_1$ to be the trivial subspace ${0}$, $V_2$ to be the entire space $V$, and $U$ to be $V$. Then $V_1 + U = {0} + V = V$ and $V_2 + U = V + V = V$, so $V_1 + U = V_2 + U$. However, $V_1 \neq V_2$ unless $V$ is the zero vector space. Intuitively, this counterexample works because adding the same subspace $U$ to different subspaces $V_1$ and $V_2$ can "hide" the differences between $V_1$ and $V_2$. In this case, adding $V$ to ${0}$ and to $V$ yields the same result, namely $V$, despite ${0}$ and $V$ being different. This exercise highlights the importance of being cautious when cancelling or simplifying equations involving subspaces. While certain cancellation laws hold for subspaces (for instance, if $U_1 + W = U_2 + W$ and $U_1, U_2, W$ are subspaces with $U_1 \subseteq W$ and $U_2 \subseteq W$, then $U_1 = U_2$), they don't hold in general, as this exercise demonstrates. $\textit{Example:}$ In $\mathbb{R}^3$, let $V_1 = {0}$, $V_2$ be the $xy$-plane, and $U = \mathbb{R}^3$. Then $V_1 + U = V_2 + U = \mathbb{R}^3$, but $V_1 \neq V_2$. In $\mathbb{F}_p^2$, let $V_1 = {(0,0)}$, $V_2 = {(x,0) : x \in \mathbb{F}_p}$, and $U = \mathbb{F}_p^2$. Then $V_1 + U = V_2 + U = \mathbb{F}_p^2$, but $V_1 \neq V_2$. In the space of polynomials $\mathbb{Q}[x]$, let $V_1 = {0}$, $V_2$ be the subspace of constant polynomials, and $U = \mathbb{Q}[x]$. Then $V_1 + U = V_2 + U = \mathbb{Q}[x]$, but $V_1 \neq V_2$. These examples show that the phenomenon of different subspaces yielding the same sum with another subspace is not limited to any particular type of vector space or field. It can occur in finite-dimensional spaces over any field, as well as in infinite-dimensional spaces like polynomial spaces.