Suppose T ∈ L(V ) and U is a subspace of V . (a) Prove that if U ⊆ null T, then U is invariant under T. (b) Prove
that if range T ⊆ U, then U is invariant under T.

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.

Suppose T ∈ L(V ) and U is a subspace of V . (a) Prove that if U ⊆ null T, then U is invariant under T. (b) Prove that if range T ⊆ U, then U is invariant under T.

Learn how to prove that a subspace U is invariant under a linear transformation T. This detailed solution explores the relationship between null space, range, and subspaces, focusing on key concepts in linear algebra. Suitable for undergraduate math students.

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