Here are the exercise problems converted to LaTeX: 1C Exercise 1 \begin{exercise} For each of the following subsets of $\mathbb{F}^3$, determine whether it is a subspace of $\mathbb{F}^3$. \begin{enumerate}[(a)] \item $\{(x_1,x_2,x_3) \in \mathbb{F}^3 : x_1 + 2x_2 + 3x_3 = 0\}$ \item $\{(x_1,x_2,x_3) \in \mathbb{F}^3 : x_1 + 2x_2 + 3x_3 = 4\}$ \item $\{(x_1,x_2,x_3) \in \mathbb{F}^3 : x_1x_2x_3 = 0\}$ \item $\{(x_1,x_2,x_3) \in \mathbb{F}^3 : x_1 = 5x_3\}$ \end{enumerate} \end{exercise} 1C Exercise 2 \begin{exercise} Verify all assertions about subspaces in Example 1.35. \end{exercise} 1C Exercise 3 \begin{exercise} Show that the set of differentiable real-valued functions $f$ on the interval $(-4,4)$ such that $f'(-1) = 3f(2)$ is a subspace of $\mathbb{R}^{(-4,4)}$. \end{exercise} 1C Exercise 4 \begin{exercise} Suppose $b \in \mathbb{R}$. Show that the set of continuous real-valued functions $f$ on the interval $[0,1]$ such that $\int_0^1 f = b$ is a subspace of $\mathbb{R}^{[0,1]}$ if and only if $b=0$. \end{exercise} 1C Exercise 5 \begin{exercise} Is $\mathbb{R}^2$ a subspace of the complex vector space $\mathbb{C}^2$? \end{exercise} 1C Exercise 6 \begin{exercise} \begin{enumerate}[(a)] \item Is $\{(a,b,c) \in \mathbb{R}^3 : a^3 = b^3\}$ a subspace of $\mathbb{R}^3$? \item Is $\{(a,b,c) \in \mathbb{C}^3 : a^3 = b^3\}$ a subspace of $\mathbb{C}^3$? \end{enumerate} \end{exercise} 1C Exercise 7 \begin{exercise} Prove or give a counterexample: If $U$ is a nonempty subset of $\mathbb{R}^2$ such that $U$ is closed under addition and under taking additive inverses (meaning $-\mathbf{u} \in U$ whenever $\mathbf{u} \in U$), then $U$ is a subspace of $\mathbb{R}^2$. \end{exercise} 1C Exercise 8 \begin{exercise} Give an example of a nonempty subset $U$ of $\mathbb{R}^2$ such that $U$ is closed under scalar multiplication, but $U$ is not a subspace of $\mathbb{R}^2$. \end{exercise} 1C Exercise 9 \begin{exercise} A function $f \colon \mathbb{R} \to \mathbb{R}$ is called \textbf{periodic} if there exists a positive number $p$ such that $f(x) = f(x+p)$ for all $x \in \mathbb{R}$. Is the set of periodic functions from $\mathbb{R}$ to $\mathbb{R}$ a subspace of $\mathbb{R}^\mathbb{R}$? Explain. \end{exercise} 1C Exercise 10 \begin{exercise} Suppose $V_1$ and $V_2$ are subspaces of $V$. Prove that the intersection $V_1 \cap V_2$ is a subspace of $V$. \end{exercise} 1C Exercise 11 \begin{exercise} Prove that the intersection of every collection of subspaces of $V$ is a subspace of $V$. \end{exercise} 1C Exercise 12 \begin{exercise} Prove that the union of two subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces is contained in the other. \end{exercise} 1C Exercise 13 \begin{exercise} Prove that the union of three subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces contains the other two. This exercise is surprisingly harder than Exercise 12, possibly because this exercise is not true if we replace $\mathbb{F}$ with a field containing only two elements. \end{exercise} 1C Exercise 14 \begin{exercise} Suppose $U = \{(x,-x,2x) \in \mathbb{F}^3 : x \in \mathbb{F}\}$ and $W = \{(x,x,2x) \in \mathbb{F}^3 : x \in \mathbb{F}\}$. Describe $U+W$ using symbols, and also give a description of $U+W$ that uses no symbols. \end{exercise} 1C Exercise 15 \begin{exercise} Suppose $U$ is a subspace of $V$. What is $U+U$? \end{exercise} 1C Exercise 16 \begin{exercise} Is the operation of addition on the subspaces of $V$ commutative? In other words, if $U$ and $W$ are subspaces of $V$, is $U+W = W+U$? \end{exercise} 1C Exercise 17 \begin{exercise} Is the operation of addition on the subspaces of $V$ associative? In other words, if $V_1,V_2,V_3$ are subspaces of $V$, is $(V_1+V_2)+V_3 = V_1+(V_2+V_3)$? \end{exercise} 1C Exercise 18 \begin{exercise} Does the operation of addition on the subspaces of $V$ have an additive identity? Which subspaces have additive inverses? \end{exercise} 1C Exercise 19 \begin{exercise} Prove or give a counterexample: If $V_1,V_2,U$ are subspaces of $V$ such that $V_1+U = V_2+U$, then $V_1 = V_2$. \end{exercise} 1C Exercise 20 \begin{exercise} 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 $\mathbb{F}^4 = U \oplus W$. \end{exercise} 1C Exercise 21 \begin{exercise} 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 $\mathbb{F}^5 = U \oplus W$. \end{exercise} 1C Exercise 22 \begin{exercise} Suppose $U = \{(x,y,x+y,x-y,2x) \in \mathbb{F}^5 : x,y \in \mathbb{F}\}$. Find three subspaces $W_1,W_2,W_3$ of $\mathbb{F}^5$, none of which equals $\{\mathbf{0}\}$, such that $\mathbb{F}^5 = U \oplus W_1 \oplus W_2 \oplus W_3$. \end{exercise} 1C Exercise 23 \begin{exercise} Prove or give a counterexample: If $V_1,V_2,U$ are subspaces of $V$ such that $V = V_1 \oplus U$ and $V = V_2 \oplus U$, then $V_1 = V_2$. Hint: When trying to discover whether a conjecture in linear algebra is true or false, it is often useful to start by experimenting in $\mathbb{F}^2$. \end{exercise} 1C Exercise 24 \begin{exercise} A function $f \colon \mathbb{R} \to \mathbb{R}$ is called \textbf{even} if $f(-x) = f(x)$ for all $x \in \mathbb{R}$. A function $f \colon \mathbb{R} \to \mathbb{R}$ is called \textbf{odd} if $f(-x) = -f(x)$ for all $x \in \mathbb{R}$. Let $V_e$ denote the set of real-valued even functions on $\mathbb{R}$ and let $V_o$ denote the set of real-valued odd functions on $\mathbb{R}$. Show that $\mathbb{R}^\mathbb{R} = V_e \oplus V_o$. \end{exercise}