To determine if the vectors $v = \begin{bmatrix} 1 \\ -5 \\ -2 \end{bmatrix}$ and $v = \begin{bmatrix} 1 \\ -5 \\ 1 \end{bmatrix}$ are solutions to the system, we should analyze if they can be expressed as a linear combination of the given vectors $v_1 = \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}$ and $v_2 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$, which are already solutions to the homogeneous linear system.

The problem is set in the context of a homogeneous system with four equations and three unknowns, meaning that any vector that can be represented as a linear combination of $v_1$ and $v_2$ will also satisfy the system.

Let's proceed by expressing each target vector as a potential linear combination of $v_1$ and $v_2$:

1. For the vector $v = \begin{bmatrix} 1 \\ -5 \\ -2 \end{bmatrix}$: We assume: $\begin{bmatrix} 1 \\ -5 \\ -2 \end{bmatrix} = a \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} + b \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ Expanding this, we get the following system of equations: $a + b = 1$ $-a + b = -5$ $b = -2$ Solving this system, we find values for $a$ and $b$. If consistent, $v = \begin{bmatrix} 1 \\ -5 \\ -2 \end{bmatrix}$ can be expressed as a combination of $v_1$ and $v_2$.

2. For the vector $v = \begin{bmatrix} 1 \\ -5 \\ 1 \end{bmatrix}$: Similarly, we assume: $\begin{bmatrix} 1 \\ -5 \\ 1 \end{bmatrix} = a \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} + b \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ Expanding, we derive a similar system of equations.