Invariant Subspaces Examples 1

# 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 $T(u) \in U$.