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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.

Given a homogeneous linear system over ℝ with four equations and three unknowns, where two vectors \( v_1 = \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} \) and \( v_2 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \) are solutions to the system. Determine if the vectors \( v = \begin{bmatrix} 1 \\ -5 \\ -2 \end{bmatrix} \) and \( v = \begin{bmatrix} 1 \\ -5 \\ 1 \end{bmatrix} \) are also solutions to the system.

Explore how to determine if given vectors are solutions to a homogeneous linear system with four equations and three unknowns. This example includes key linear algebra concepts such as vector linear combinations and spans, suitable for university-level students.

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