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.