To determine if b is a linear combination of a₁, a₂, and a₃, we need to check if there exist scalars $x_1, x_2, x_3$ such that:

$x_1 \cdot a_1 + x_2 \cdot a_2 + x_3 \cdot a_3 = b$

This can be written as the following matrix equation:

$x_1 \begin{bmatrix} 1 \\ -2 \\ 0 \end{bmatrix} + x_2 \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} + x_3 \begin{bmatrix} 5 \\ -6 \\ 8 \end{bmatrix} = \begin{bmatrix} 2 \\ -1 \\ 6 \end{bmatrix}$

This leads to the system of linear equations:

1. & \quad x_1 + 0x_2 + 5x_3 = 2 \\ 2. & \quad -2x_1 + x_2 - 6x_3 = -1 \\ 3. & \quad 0x_1 + 2x_2 + 8x_3 = 6 \end{aligned}$$ We will solve this system to determine whether a solution exists for $$x_1, x_2, x_3$$. Let me solve it.The solution to the system is given by: $$x_1 = 2 - 5x_3, \quad x_2 = 3 - 4x_3$$ This means that there are infinitely many solutions depending on the value of $$x_3$$. Therefore, **b** is a linear combination of $$a_1$$, $$a_2$$, and $$a_3$$, and the scalars $$x_1$$, $$x_2$$, and $$x_3$$ are not unique, but dependent on the value of $$x_3$$. Would you like more details, or do you have any questions? Here are five related questions you could explore: 1. How do you express the general form of linear combinations? 2. What is the geometric interpretation of a linear combination of vectors? 3. How can you determine if a set of vectors is linearly dependent? 4. What does it mean when a system of linear equations has infinitely many solutions? 5. Can you find a specific solution by assigning a value to $$x_3$$? **Tip:** When a system of equations has infinitely many solutions, it means the vectors involved are not linearly independent!