Let's break down each part of the problem.

(a) Prove that if $U \subseteq \text{null}(T)$, then $U$ is invariant under $T$.

Proof:

Let $T \in L(V)$, where $L(V)$ is the set of all linear operators on $V$, and let $U$ be a subspace of $V$ such that $U \subseteq \text{null}(T)$. We want to show that $U$ is invariant under $T$, i.e., for every $u \in U$, $T(u) \in U$.

Since $U \subseteq \text{null}(T)$, by definition of the null space, for every $u \in U$, we have: $T(u) = 0.$ Thus, for every $u \in U$, $T(u) = 0$, which implies that $T(u) \in U$ (since $0 \in U$, because $U$ is a subspace).

Therefore, $U$ is invariant under $T$.

(b) Prove that if $\text{range}(T) \subseteq U$, then $U$ is invariant under $T$.

Proof:

Let $T \in L(V)$, where $L(V)$ is the set of all linear operators on $V$, and let $U$ be a subspace of $V$ such that $\text{range}(T) \subseteq U$. We want to show that $U$ is invariant under $T$, i.e., for every $u \in U$, $T(u) \in U$.

Since $\text{range}(T) \subseteq U$, for every $v \in V$, we have: $T(v) \in U.$ In particular, for every $u \in U$, we know that $T(u) \in \text{range}(T)$ by the definition of a linear transformation. Since $\text{range}(T) \subseteq U$, it follows that: $T(u) \in U.$ Thus, $U$ is invariant under $T$.

Conclusion:

• For part (a), we showed that if $U \subseteq \text{null}(T)$, then $T(u) = 0$ for every $u \in U$, which implies that $U$ is invariant under $T$.
• For part (b), we showed that if $\text{range}(T) \subseteq U$, then for every $u \in U$, $T(u) \in U$, which also implies that $U$ is invariant under $T$.

Do you have any questions or would you like further details?

Here are 5 related questions for further exploration:

1. What is the relationship between the null space and the range of a linear operator?
2. How do invariant subspaces relate to the diagonalizability of a linear operator?
3. What are examples of operators where every subspace is invariant?
4. How do the kernel and image of a linear transformation help in finding invariant subspaces?
5. Can a subspace be invariant under two different linear operators simultaneously?

Tip: When proving properties of linear transformations, it's helpful to use the definitions of the null space and range, and verify how each element behaves under the transformation.