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.