Suppose 𝑆, 𝑇 ∈ ℒ(𝑉) are such that range 𝑆 ⊆ null 𝑇. Prove that (𝑆𝑇)^2 = 0

Given that \( S, T \in \mathcal{L}(V) \) are linear operators on a vector space \( V \), and the range of \( S \) is contained in the null space of \( T \) (i.e., \(\text{range}(S) \subseteq \text{null}(T)\)), we want to prove that \( (ST)^2 = 0 \), or equivalently \( STST = 0 \).

### Proof:

1. **Understanding the Given Condition**:
   - The condition \( \text{range}(S) \subseteq \text{null}(T) \) implies that for every vector \( v \in V \), the vector \( S(v) \) (which belongs to the range of \( S \)) is also in the null space of \( T \).
   - Therefore, \( T(S(v)) = 0 \) for all \( v \in V \).

2. **Expression of \( (ST)^2 \)**:
   - We need to prove that \( (ST)^2 = 0 \). By definition:
     \[
     (ST)^2 = STST
     \]
   - We will show that for any \( v \in V \), \( STST(v) = 0 \).

3. **Application of the Given Condition**:
   - Consider \( ST(v) \) for any \( v \in V \). Since \( S(v) \in V \), applying \( T \) to \( S(v) \) gives us:
     \[
     T(S(v)) = 0
     \]
   - This directly implies that:
     \[
     ST(v) \in \text{range}(S) \subseteq \text{null}(T)
     \]
   - Thus, applying \( T \) again to \( ST(v) \), we get:
     \[
     T(ST(v)) = 0
     \]
   - Therefore:
     \[
     STST(v) = S(T(ST(v))) = S(0) = 0
     \]

4. **Conclusion**:
   - Since \( STST(v) = 0 \) for any vector \( v \in V \), it follows that \( STST = 0 \), which is the same as saying \( (ST)^2 = 0 \).

Hence, we have proved that \( (ST)^2 = 0 \).

Would you like more details or have any further questions?

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Here are five related questions for further exploration:
1. What are the implications of \( S \) and \( T \) being nilpotent operators?
2. How would the result change if we assumed that \( \text{null}(S) \subseteq \text{range}(T) \)?
3. Can we generalize this result to any linear operators \( S \) and \( T \) with similar range and null space conditions?
4. What is the significance of the result in terms of the minimal polynomial of the operator \( ST \)?
5. How does the dimension of \( V \) affect the properties of \( S \) and \( T \)?

**Tip**: When dealing with linear operators, carefully analyze the relationships between their range and null spaces to uncover deeper properties of the operators.

Learn how to prove that the square of the product of linear operators S and T, where the range of S is contained in the null space of T, equals zero. This problem involves advanced concepts in linear algebra and vector spaces.

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