Suppose S, T is in L(V) are such that range S is a subspace of null T. Prove that (ST)^2 = 0

To prove that \((ST)^2 = 0\), we need to use the given information that \( \text{range}(S) \subseteq \text{null}(T) \). We will proceed step-by-step:

### Step 1: Definition of \((ST)^2\)
We need to show that \((ST)^2 = 0\). This expression expands as:
\[
(ST)^2 = STST
\]
So, we need to show that \( STST = 0 \) for any vector \( v \in V \).

### Step 2: Using the assumption about the range of \(S\)
From the assumption \( \text{range}(S) \subseteq \text{null}(T) \), we know that for any vector \( v \in V \), the vector \( S(v) \) is in the null space of \(T\). This means that:
\[
T(S(v)) = 0 \quad \text{for all} \quad v \in V.
\]
In other words, applying \(T\) to any vector in the range of \(S\) gives 0.

### Step 3: Applying \( STST \) to an arbitrary vector \(v\)
Now, let's compute \( STST(v) \) for an arbitrary vector \(v \in V\):
\[
STST(v) = ST(S(T(v))).
\]
First, apply \( T \) to \( v \), which gives \( T(v) \in V \). Then apply \( S \) to \( T(v) \), which gives \( S(T(v)) \).

Now, since \( S(T(v)) \) is in the range of \( S \) and by our assumption \( \text{range}(S) \subseteq \text{null}(T) \), we have:
\[
T(S(T(v))) = 0.
\]

Thus,
\[
STST(v) = S(0) = 0.
\]

### Step 4: Conclusion
Since \( STST(v) = 0 \) for all \( v \in V \), it follows that:
\[
(ST)^2 = 0.
\]

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

---

Would you like further clarification on any step, or details on how these concepts apply in linear algebra?

Here are 5 related questions to deepen your understanding:
1. What is the null space of a linear transformation?
2. Can you explain the relationship between a linear map's kernel and image?
3. How does the property \( \text{range}(S) \subseteq \text{null}(T) \) affect the behavior of the product \(ST\)?
4. What is the significance of nilpotent operators in linear algebra?
5. How would the proof change if \( \text{range}(S) \) were not a subspace of \( \text{null}(T) \)?

**Tip**: The null space (or kernel) of a linear transformation consists of all vectors that the transformation maps to the zero vector. Understanding its geometric and algebraic properties is key to solving many linear algebra problems.

Suppose S, T is in L(V) are such that range S is a subspace of null T. Prove that (ST)^2 = 0.

Explore the step-by-step proof that (ST)^2 = 0, given that the range of S is a subspace of the null space of T. This problem involves core linear algebra concepts such as null spaces, ranges, and linear transformations, making it ideal for undergraduate students.

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