The problem asks you to express the vector w as a linear combination of the vectors v₁ and v₂.

Given: $\mathbf{w} = \begin{bmatrix} -8 \\ 11 \\ 8 \end{bmatrix}, \quad \mathbf{v_1} = \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{v_2} = \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}$

We need to find scalars $c_1$ and $c_2$ such that: $\mathbf{w} = c_1 \mathbf{v_1} + c_2 \mathbf{v_2}$ This can be written as the system of equations: $\begin{bmatrix} -8 \\ 11 \\ 8 \end{bmatrix} = c_1 \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}$ This gives the following system of equations:

1. $-c_1 + 2c_2 = -8$
2. $c_1 + c_2 = 11$
3. $c_1 + c_2 = 8$

However, notice that equations 2 and 3 are contradictory, because both cannot be true simultaneously. This means there is no solution for $c_1$ and $c_2$ that satisfies all three equations.

Therefore, the vector w cannot be expressed as a linear combination of v₁ and v₂, and the answer is DNE (Does Not Exist).

Would you like further details or have any questions about this? Here are 5 related questions for further exploration:

1. How do you solve a system of equations for linear combinations?
2. What does it mean if a vector cannot be expressed as a linear combination of other vectors?
3. What is the geometric interpretation of a linear combination in 3D space?
4. How would you check if a set of vectors is linearly independent?
5. What is the rank of a matrix, and how is it related to linear combinations?

Tip: When equations in a system contradict each other, it typically means the vectors are linearly independent and the combination does not exist.