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The image contains a problem about analyzing a set of vectors to check for linear dependence. Here’s a step-by-step explanation of the process presented in the image:

### Problem Statement:
- **Given Vectors:** \( \mathbf{v}_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix} \), \( \mathbf{v}_2 = \begin{pmatrix} 3 \\ 6 \\ 9 \\ 12 \end{pmatrix} \), \( \mathbf{v}_3 = \begin{pmatrix} 1 \\ -2 \\ 1 \\ -2 \end{pmatrix} \)

### Step 1: Analyze Set (a)
#### **Check for Linear Dependence:**

1. **Compare the first two vectors:**  
   \( \mathbf{v}_2 = \begin{pmatrix} 3 \\ 6 \\ 9 \\ 12 \end{pmatrix} \) and \( \mathbf{v}_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix} \).
   
2. **Identify if one vector is a scalar multiple of another:**
   - Notice that \( \mathbf{v}_2 = 3 \times \mathbf{v}_1 \).
   
3. **Conclusion:**
   - Since one vector is a scalar multiple of another, the set is linearly dependent.

Therefore, the set \( \{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \) is linearly dependent.

### Next Steps or Details:
Would you like to go through further details, such as exploring what linear dependence means in the context of vector spaces or looking at the implications of this result?

### Questions to Expand Your Understanding:
1. What does it mean for a set of vectors to be linearly dependent?
2. How would you determine if three vectors in \(\mathbb{R}^3\) are linearly dependent?
3. Can you explain what it means for one vector to be a scalar multiple of another?
4. What is the significance of linear dependence in solving systems of equations?
5. How would the analysis differ if the vectors were in a higher-dimensional space like \(\mathbb{R}^4\)?

### Tip:
When checking for linear dependence, always look for whether one vector can be expressed as a combination of others; this often involves simple scalar multiples or linear combinations.

Learn how to analyze a set of vectors for linear dependence with a step-by-step explanation. This problem involves vectors in \(\mathbb{R}^4\), checking for scalar multiples, and understanding linear dependence in vector spaces.

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