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The problem in the image is asking to determine which combination of vectors \( \alpha_1, \alpha_2, \alpha_3, \alpha_4 \) forms a linearly dependent set.

Given the options:

A. \( \alpha_1 - \alpha_2, \alpha_2 + \alpha_3, \alpha_3 + \alpha_4, \alpha_4 + \alpha_1 \)  
B. \( \alpha_1 + \alpha_2, \alpha_2 + \alpha_3, \alpha_3 + \alpha_4, \alpha_4 - \alpha_1 \)  
C. \( \alpha_1 + \alpha_2, \alpha_2 + \alpha_3, \alpha_3 + \alpha_4, \alpha_4 + \alpha_1 \)  
D. \( \alpha_1 + \alpha_2, \alpha_3 - \alpha_2, \alpha_3 + \alpha_4, \alpha_4 - \alpha_1 \)  

### Solution:

For a set of vectors to be linearly dependent, there must exist a non-trivial combination of these vectors that sums to zero.

1. **Option A:**  
   Sum of the vectors:  
   \[
   (\alpha_1 - \alpha_2) + (\alpha_2 + \alpha_3) + (\alpha_3 + \alpha_4) + (\alpha_4 + \alpha_1)
   \]
   Simplifies to:
   \[
   2\alpha_1 + 2\alpha_3 + 2\alpha_4
   \]
   This combination isn't zero unless \(\alpha_1, \alpha_3, \alpha_4\) are specifically chosen.

2. **Option B:**  
   Sum of the vectors:  
   \[
   (\alpha_1 + \alpha_2) + (\alpha_2 + \alpha_3) + (\alpha_3 + \alpha_4) + (\alpha_4 - \alpha_1)
   \]
   Simplifies to:
   \[
   2\alpha_2 + 2\alpha_3 + 2\alpha_4
   \]
   This combination isn't zero unless \(\alpha_2, \alpha_3, \alpha_4\) are specifically chosen.

3. **Option C:**  
   Sum of the vectors:  
   \[
   (\alpha_1 + \alpha_2) + (\alpha_2 + \alpha_3) + (\alpha_3 + \alpha_4) + (\alpha_4 + \alpha_1)
   \]
   Simplifies to:
   \[
   2(\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4)
   \]
   This is linearly dependent since the sum equals zero if \( \alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 = 0 \).

4. **Option D:**  
   Sum of the vectors:  
   \[
   (\alpha_1 + \alpha_2) + (\alpha_3 - \alpha_2) + (\alpha_3 + \alpha_4) + (\alpha_4 - \alpha_1)
   \]
   Simplifies to:
   \[
   2\alpha_3 + 2\alpha_4
   \]
   This combination isn't zero unless \(\alpha_3, \alpha_4\) are specifically chosen.

### Conclusion:
The correct answer is **C** because the set of vectors forms a linearly dependent set under the condition \( \alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 = 0 \).

Would you like further details, or do you have any other questions?

Here are five additional related questions:

1. What is the definition of linear dependence and independence in vectors?
2. How do you determine the rank of a matrix in relation to linear dependence?
3. Can you explain the relationship between the span of vectors and their linear dependence?
4. How does the concept of linear dependence apply to systems of linear equations?
5. What are the implications of linear dependence in the context of vector spaces?

**Tip:** When solving problems involving linear dependence, always check if you can form a non-trivial combination of vectors that results in the zero vector.

Explore how to identify linear dependence in vectors with a detailed analysis of a problem involving vector combinations. Understand the conditions under which vectors form a linearly dependent set. Suitable for advanced high school students and learners interested in linear algebra concepts.

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