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Let's break down and solve the given problem step by step.

We are given:

- \( U, V, W \) are vector spaces.
- \( T: U \to V \) and \( S: V \to W \) are linear maps.
- We need to prove that \( \ker(T) \subseteq \ker(S \circ T) \), where \( S \circ T: U \to W \) is the composition of \( S \) and \( T \).

### Proof:

1. **Recall the definition of the kernel**:
   - The kernel of a linear map \( T: U \to V \), denoted \( \ker(T) \), is the set of all vectors \( u \in U \) such that \( T(u) = 0 \).
   - Similarly, \( \ker(S \circ T) \) is the set of vectors \( u \in U \) such that \( (S \circ T)(u) = 0 \).

2. **Start with \( u \in \ker(T) \)**:
   - By definition of the kernel, if \( u \in \ker(T) \), then \( T(u) = 0 \).

3. **Apply the composition \( S \circ T \)**:
   - Now consider \( (S \circ T)(u) = S(T(u)) \).
   - Since \( T(u) = 0 \), this becomes \( S(0) \).

4. **Use the fact that \( S \) is linear**:
   - Because \( S \) is a linear map, we know that \( S(0) = 0 \).
   - Therefore, \( (S \circ T)(u) = 0 \).

5. **Conclusion**:
   - We have shown that if \( u \in \ker(T) \), then \( u \in \ker(S \circ T) \).
   - This proves that \( \ker(T) \subseteq \ker(S \circ T) \), as required.

Thus, the proof is complete.

Would you like further details or have any questions?

Here are five related questions:
1. What is the composition of two linear maps?
2. How can the rank-nullity theorem apply to this situation?
3. Can we generalize this result to non-linear maps?
4. How does the inclusion \( \ker(T) \subseteq \ker(S \circ T) \) relate to injectivity of \( T \)?
5. How would the proof change if \( S \) were not linear?

**Tip:** When proving set inclusions, always begin by assuming an element belongs to one set and show that it must belong to the other.

Suppose U, V, W are vector spaces, and T: U → V and S: V → W are linear functions. Prove that ker(T) ⊆ ker(S ∘ T).

Learn how to prove that ker(T) ⊆ ker(S ∘ T) where T: U → V and S: V → W are linear maps between vector spaces U, V, and W. This solution explores core concepts from linear algebra such as the kernel of a linear map, and the composition of linear maps. Ideal for students studying undergraduate-level linear algebra.

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