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.