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The problem you're asking about, Exercise 3, presents three vectors:

\[
\mathbf{v_1} = \begin{bmatrix} 1 \\ -3 \\ 9 \end{bmatrix}, \quad \mathbf{v_2} = \begin{bmatrix} 1 \\ -3 \\ 9 \end{bmatrix}, \quad \mathbf{v_3} = \begin{bmatrix} 1 \\ -3 \\ 9 \end{bmatrix}
\]

The task is to determine whether these vectors are **linearly independent**.

### Definition of Linear Independence:
Vectors are said to be linearly independent if no vector in the set can be written as a linear combination of the others. Mathematically, for vectors \(\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}\), they are linearly independent if the only solution to the equation:

\[
c_1 \mathbf{v_1} + c_2 \mathbf{v_2} + c_3 \mathbf{v_3} = \mathbf{0}
\]

is \(c_1 = c_2 = c_3 = 0\). If any non-trivial solution exists (i.e., \(c_1, c_2, c_3\) are not all zero), then the vectors are linearly dependent.

### Solving the Problem:
All three vectors given here are identical. Therefore, they are not linearly independent. In fact, any two identical vectors are automatically linearly dependent because one can be written as a scalar multiple of the other. Specifically, for this set of vectors, we can write:

\[
\mathbf{v_1} = 1 \cdot \mathbf{v_2}, \quad \mathbf{v_2} = 1 \cdot \mathbf{v_3}
\]

Thus, there exist non-trivial scalars that satisfy the equation \(c_1 \mathbf{v_1} + c_2 \mathbf{v_2} + c_3 \mathbf{v_3} = \mathbf{0}\), meaning the vectors are **linearly dependent**.

### Conclusion:
The set of vectors \(\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}\) are **linearly dependent**.

Would you like further clarification or details on this solution?

Here are some related questions to deepen understanding:
1. What conditions guarantee linear independence for any set of vectors?
2. How can the rank of a matrix help determine the linear dependence of its columns?
3. What is the geometric interpretation of linear independence in \(\mathbb{R}^3\)?
4. How would you use row reduction to determine linear dependence of vectors?
5. Can you explain how linear dependence relates to the concept of a span?

**Tip:** When analyzing linear dependence or independence, checking if the determinant of the matrix formed by vectors is zero can help with small sets of vectors.

Determine if the vectors v1 = [1 -3 9], v2 = [1 -3 9], and v3 = [1 -3 9] are linearly independent or dependent. Justify your answer.

Solve a linear algebra problem determining whether the vectors v1 = [1 -3 9], v2 = [1 -3 9], and v3 = [1 -3 9] are linearly dependent or independent. This exercise covers key concepts in vector spaces, including linear independence, and is suitable for undergraduate students.

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