To determine if the vectors $\mathbf{v}_1 = \begin{bmatrix} -2 \\ 4 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} 7 \\ -2 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix} 3 \\ -6 \end{bmatrix}$ are linearly independent, we check if there exists a non-trivial solution to the following equation:

$c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + c_3 \mathbf{v}_3 = \mathbf{0}$

This translates to:

$c_1 \begin{bmatrix} -2 \\ 4 \end{bmatrix} + c_2 \begin{bmatrix} 7 \\ -2 \end{bmatrix} + c_3 \begin{bmatrix} 3 \\ -6 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$

We now have two equations:

$-2c_1 + 7c_2 + 3c_3 = 0$ $4c_1 - 2c_2 - 6c_3 = 0$

Solving this system of equations will tell us if the only solution is $c_1 = c_2 = c_3 = 0$ (indicating linear independence) or if a non-trivial solution exists (indicating linear dependence).

Let me solve these equations for you.

Step 1: Simplify the second equation

Divide the second equation by 2:

$2c_1 - c_2 - 3c_3 = 0$

Now, the system is:

$-2c_1 + 7c_2 + 3c_3 = 0 \quad \text{(1)}$ $2c_1 - c_2 - 3c_3 = 0 \quad \text{(2)}$

Step 2: Solve the system

We can solve this system of equations for $c_1, c_2, c_3$. I'll proceed with this calculation.

The solution to the system is:

$c_1 = \frac{3}{2}c_3, \quad c_2 = 0$

This shows that the vectors are linearly dependent, because there exists a non-trivial solution where $c_3 \neq 0$ leads to $c_1 \neq 0$. Hence, the vectors $\mathbf{v}_1$, $\mathbf{v}_2$, and $\mathbf{v}_3$ are linearly dependent.

Would you like further clarification or details on this process?

Related Questions:

1. What is the geometric meaning of linear dependence for 2D vectors?
2. How do we interpret the solution $c_1 = \frac{3}{2}c_3$ in terms of vector relations?
3. Can a set of two vectors ever be linearly independent in 2D space?
4. How would you extend this method to check for linear independence in three dimensions?
5. How does the determinant of a matrix relate to linear independence?

Tip: When checking for linear independence of multiple vectors, solving the system of equations helps reveal any non-trivial relationships between the vectors.