$\textbf{Exercise 1.}$ Suppose $T \in \mathcal{L}(V)$ and $U$ is a subspace of $V$. (a) Prove that if $U \subseteq \operatorname{null} T$, then $U$ is invariant under $T$. (b) Prove that if $\operatorname{range} T \subseteq U$, then $U$ is invariant under $T$. $\textbf{Solution 1.}$ (a) Suppose $U \subseteq \operatorname{null} T$. If $u \in U$, then $u \in \operatorname{null} T$, and hence $Tu = 0$, and thus $Tu \in U$. Thus $U$ is invariant under $T$. (b) Suppose $\operatorname{range} T \subseteq U$. If $u \in U$, then $Tu \in \operatorname{range} T$, and hence $Tu \in U$. Thus $U$ is invariant under $T$. ---  --- Commentary: $\textbf{Exercise 1.}$ Suppose $T \in \mathcal{L}(V)$ and $U$ is a subspace of $V$. (a) Prove that if $U \subseteq \operatorname{null} T$, then $U$ is invariant under $T$. (b) Prove that if $\operatorname{range} T \subseteq U$, then $U$ is invariant under $T$. $\textbf{Solution 1.}$ (a) Suppose $U \subseteq \operatorname{null} T$. If $u \in U$, then $u \in \operatorname{null} T$, and hence $Tu = 0$, and thus $Tu \in U$. Thus $U$ is invariant under $T$. (b) Suppose $\operatorname{range} T \subseteq U$. If $u \in U$, then $Tu \in \operatorname{range} T$, and hence $Tu \in U$. Thus $U$ is invariant under $T$. $\textit{Commentary:}$ This exercise explores two simple conditions that ensure a subspace is invariant under a linear operator. A subspace $U$ is invariant under $T$ if $Tu \in U$ for all $u \in U$. Part (a) shows that if $U$ is contained in the null space of $T$, then it is invariant, because $Tu = 0 \in U$ for all $u \in U$. Part (b) shows that if the range of $T$ is contained in $U$, then $U$ is invariant, because $Tu \in \operatorname{range} T \subseteq U$ for all $u \in U$. These conditions are not necessary for invariance, but they are easy to check and are often useful in applications. $\textit{Example:}$ Let $V = \mathbb{R}^3$ and define $T: \mathbb{R}^3 \to \mathbb{R}^3$ by $T(x, y, z) = (0, x, 0)$. The null space of $T$ is $\operatorname{null} T = \{(0, y, z) : y, z \in \mathbb{R}\}$, the $yz$-plane. Any subspace of the $yz$-plane, such as the $y$-axis $\{(0, y, 0) : y \in \mathbb{R}\}$ or the $z$-axis $\{(0, 0, z) : z \in \mathbb{R}\}$, is invariant under $T$ by part (a). The range of $T$ is $\operatorname{range} T = \{(0, x, 0) : x \in \mathbb{R}\}$, the $x$-axis. Any subspace containing the $x$-axis, such as the $xy$-plane $\{(x, y, 0) : x, y \in \mathbb{R}\}$, is invariant under $T$ by part (b).