We are asked to express the vector $\mathbf{v} = (-4, 7, 1)$ as a linear combination of the vectors $\mathbf{v_1} = (0, 1, 1)$, $\mathbf{v_2} = (1, 2, -2)$, and $\mathbf{v_3} = (3, 1, 0)$.

In other words, we need to find scalars $a$, $b$, and $c$ such that: $\mathbf{v} = a \mathbf{v_1} + b \mathbf{v_2} + c \mathbf{v_3}$ This can be written as the following system of equations based on the components of each vector:

$a(0, 1, 1) + b(1, 2, -2) + c(3, 1, 0) = (-4, 7, 1)$

Breaking it down into three separate equations for each component:

1. For the first component: $0a + 1b + 3c = -4$ $b + 3c = -4$

2. For the second component: $1a + 2b + 1c = 7$ $a + 2b + c = 7$

3. For the third component: $1a - 2b + 0c = 1$ $a - 2b = 1$

Solving the system of equations:

We have the following system of three equations:

1. $b + 3c = -4$
2. $a + 2b + c = 7$
3. $a - 2b = 1$

Let’s solve this step by step.

1. Solve for $b$ in terms of $c$ from equation 1: $b = -4 - 3c$

2. Substitute this expression for $b$ into equation 2: $a + 2(-4 - 3c) + c = 7$ $a - 8 - 6c + c = 7$ $a - 8 - 5c = 7$ $a = 15 + 5c$

3. Substitute this expression for $a$ and $b = -4 - 3c$ into equation 3: $(15 + 5c) - 2(-4 - 3c) = 1$ $15 + 5c + 8 + 6c = 1$ $23 + 11c = 1$ $11c = -22$ $c = -2$

4. Substitute $c = -2$ into the equation for $b$: $b = -4 - 3(-2) = -4 + 6 = 2$

5. Finally, substitute $c = -2$ into the equation for $a$: $a = 15 + 5(-2) = 15 - 10 = 5$

Final Answer:

The scalars are: $a = 5, \quad b = 2, \quad c = -2$

Thus, the vector $\mathbf{v} = (-4, 7, 1)$ can be written as: $\mathbf{v} = 5\mathbf{v_1} + 2\mathbf{v_2} - 2\mathbf{v_3}$

Would you like a more detailed breakdown of any step?

Here are five related questions:

1. How can we solve a system of linear equations using matrices?
2. What is the geometric interpretation of expressing vectors as linear combinations?
3. How do we know if a set of vectors forms a basis in $\mathbb{R}^3$?
4. What is the condition for three vectors to be linearly independent in $\mathbb{R}^3$?
5. How can linear transformations be represented using matrix operations?

Tip: When solving systems of equations, substitution works well when one variable is easy to isolate. For larger systems, consider matrix methods like Gaussian elimination.