Invariant Subspaces Examples 1
 Table of Contents

# Invariant Subspaces Examples 1

Recall from the Invariant Subspaces page that if $V$ is a vector space, $U$ is a subspace of $V$, and $T \in \mathcal L (V)$ then the subspace $U$ is said to be invariant under the linear operator $T$ if for every vector $u \in U$ we have that $T(u) \in U$.

We noted that $\{ 0 \}$, $V$, $\mathrm{null} (T)$ and $\mathrm{range} (T)$ were all invariant under $T$.

We also saw that if $V$ is a finite-dimensional vector space and $U$ is a nontrivial subspace of $V$ (that is $U$ is not the zero space $\{ 0 \}$ and not the whole space $V$) then there exists a linear operator $T \in \mathcal L(V)$ such that $U$ is not invariant under $T$.

We will now look at some examples regarding invariant subspaces.

## Example 1

Let $T \in \mathcal L(V)$ and let $U$ be a subspace of $V$. Show that if $U \subseteq \mathrm{null} (T)$ then $U$ is invariant under $T$.

We want to show that $u \in U$ implies that $T(u) \in U$.

Suppose that $U \subseteq \mathrm{null} (T)$. Then for every vector $u \in U$ we have that $T(u) = 0$. However, $0 \in U$ since $U$ is a subspace (and hence must contain the zero vector) $]] so$0 = T(u) \in U$. Therefore$u \in U$implies that$T(u) \in U$, so$U$is invariant under$T$. ## Example 2 Let$T \in \mathcal L(V)$and let$U$be a subspace of$V$. Show that if$\mathrm{range} (T) \subseteq U$then$U$is invariant under$T$. Once again, we want to show that$u \in U$implies that$T(u) \in U$. Suppose that$\mathrm{range} (T) \subseteq U$. For any vector$u \in U$we must have that$T(u) \in \mathrm{range} (T)$. However,$\mathrm{range} (T) \subseteq U$so then$T(u) \in U$as well. Therefore$u \in U$implies that$T(u) \in U$so$U$is invariant under$T$. ## Example 3 Let$T \in \mathcal L(V)$and let$U$and$W$be subspaces of$V$that are invariant under$T$. Show that$U \cap W$is invariant under$T$. We want to show that for every vector$v \in U \cap W$we have that$T(v) \in U \cap W$. Let$v \in U \cap W$. Then$v \in U$and$v \in W$. Since$U$and$W$are both invariant under$T$then we have that$T(v) \in U$and$T(v) \in W$. Therefore$T(v) \in U \cap W$. Therefore$U \cap W$is invariant under$T$. ## Example 4 Let$T \in \mathcal L(V)$and let$U_1$,$U_2$, …,$U_n$be any collection of subspaces of$V$that are invariant under$T$. Show that$\bigcap_{i=1}^{n} U_i$is invariant under$T$. We want to show that for every vector$v \in \bigcap_{i=1}^{n} U_i$we have that$T(v) \in \bigcap_{i=1}^{n} U_i$. Let$v \in \bigcap_{i=1}^{n} U_i$. Then$v \in U_1$,$v \in U_2$, …,$v \in U_n$. Since$U_1$,$U_2$, …,$U_n$are each invariant under$T$we have that$T(v) \in U_1$,$T(v) \in U_2$, …,$T(v) \in U_n$. Therefore$T(v) \in \bigcap_{i=1}^{n} U_i$and so$\bigcap_{i=1}^{n} U_i$is invariant under$T\$.

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