This problem is asking you to express the vector $\mathbf{w} = (-21, -22)$ as a linear combination of the vectors $\mathbf{v_1} = (4, 4)$, $\mathbf{v_2} = (-2, 1)$, and $\mathbf{v_3} = (1, -1)$. In other words, you need to find scalars $a$, $b$, and $c$ such that:

$\mathbf{w} = a\mathbf{v_1} + b\mathbf{v_2} + c\mathbf{v_3}$

Substituting the vectors:

$(-21, -22) = a(4, 4) + b(-2, 1) + c(1, -1)$

This gives the system of equations:

$-21 = 4a - 2b + c \quad \text{(for the x-component)}$ $-22 = 4a + b - c \quad \text{(for the y-component)}$

Now, I will solve this system for $a$, $b$, and $c$.The system of equations provides the following relationships between the scalars $a$, $b$, and $c$:

$a = \frac{c}{12} - \frac{65}{12}$ $b = \frac{2c}{3} - \frac{1}{3}$

This indicates that the solution depends on the value of $c$, as it is a free parameter. This implies that the vector $\mathbf{w}$ can be expressed as a linear combination of $\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}$, but with an infinite number of solutions, depending on the choice of $c$.

Would you like to explore any specific value for $c$, or would you like further clarification?

Here are 5 related questions you might find interesting:

1. How do you find linear independence of vectors?
2. Can any two-dimensional vector be written as a combination of three given vectors?
3. What is the significance of free parameters in systems of equations?
4. How would the solution change if the vectors were not linearly independent?
5. What methods exist to solve systems of linear equations?

Tip: In linear algebra, when solving for vector combinations, free parameters often indicate that the solution lies in a subspace.