Let's break down the problem:

You are given a homogeneous linear system over $\mathbb{R}$ with four equations and three unknowns. It is known that the vectors $\mathbf{v_1} = \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} \quad \text{and} \quad \mathbf{v_2} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ are solutions to the system.

The questions are:

1. Can we determine if the vector $\mathbf{v} = \begin{bmatrix} 1 \\ -5 \\ -2 \end{bmatrix}$ is a solution to the system?
2. Can we determine if the vector $\mathbf{v} = \begin{bmatrix} 1 \\ -5 \\ 1 \end{bmatrix}$ is a solution to the system?

Solution Approach

Since this is a homogeneous system, any solution vector $\mathbf{v}$ must be a linear combination of $\mathbf{v_1}$ and $\mathbf{v_2}$ to satisfy the system. To determine if a vector is a solution, we need to check if it can be expressed as $\mathbf{v} = a \mathbf{v_1} + b \mathbf{v_2}$ for some scalars $a$ and $b$.

1. For $\mathbf{v} = \begin{bmatrix} 1 \\ -5 \\ -2 \end{bmatrix}$:

   • Set up the equation: $a \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} + b \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ -5 \\ -2 \end{bmatrix}$
   • This gives the system: $a + b = 1$ $-a + b = -5$ $b = -2$
   • Solving this system will reveal if such $a$ and $b$ exist.

2. For $\mathbf{v} = \begin{bmatrix} 1 \\ -5 \\ 1 \end{bmatrix}$:

   • Set up the equation similarly: $a \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} + b \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ -5 \\ 1 \end{bmatrix}$
   • This gives the system: $a + b = 1$ $-a + b = -5$ $b = 1$
   • Solve this system to determine if such $a$ and $b$ exist.

Let me calculate these values for clarity.The solutions to the two questions are as follows:

1. For the vector $\mathbf{v} = \begin{bmatrix} 1 \\ -5 \\ -2 \end{bmatrix}$:

   • We found that $a = 3$ and $b = -2$ satisfy the system of equations. Therefore, this vector can be expressed as a linear combination of $\mathbf{v_1}$ and $\mathbf{v_2}$, and it is a solution to the system.

2. For the vector $\mathbf{v} = \begin{bmatrix} 1 \\ -5 \\ 1 \end{bmatrix}$:

   • There is no solution for $a$ and $b$ that satisfies the system of equations. Thus, this vector cannot be expressed as a linear combination of $\mathbf{v_1}$ and $\mathbf{v_2}$, and it is not a solution to the system.

Would you like further details or have any questions?

Here are some related questions to explore further:

1. How do you determine if a vector is a solution to a homogeneous system?
2. What are the properties of homogeneous systems in linear algebra?
3. Can a homogeneous system with more equations than unknowns have non-trivial solutions?
4. How does linear dependence between vectors affect the solution set of a system?
5. What are the implications of a solution being expressible as a linear combination of basis vectors?

Tip: In a homogeneous system, if you know some solutions, any linear combination of these solutions will also be a solution.