We are tasked with finding scalars $c_1$, $c_2$, and $c_3$ such that the vector $\mathbf{w} = \langle -1, 1, 5 \rangle$ can be written as a linear combination of the vectors $\mathbf{v_1} = \langle 1, 0, 1 \rangle$, $\mathbf{v_2} = \langle 3, 2, 0 \rangle$, and $\mathbf{v_3} = \langle 0, 1, 1 \rangle$.

This means we want to solve the following equation for $c_1$, $c_2$, and $c_3$:

$\mathbf{w} = c_1 \mathbf{v_1} + c_2 \mathbf{v_2} + c_3 \mathbf{v_3}$

Substituting the vectors, we get:

$\langle -1, 1, 5 \rangle = c_1 \langle 1, 0, 1 \rangle + c_2 \langle 3, 2, 0 \rangle + c_3 \langle 0, 1, 1 \rangle$

This expands into the following system of equations for each component:

1. $c_1(1) + c_2(3) + c_3(0) = -1$ (from the $x$-component)
2. $c_1(0) + c_2(2) + c_3(1) = 1$ (from the $y$-component)
3. $c_1(1) + c_2(0) + c_3(1) = 5$ (from the $z$-component)

We now solve this system of equations step by step:

Equation 1 (x-component):

c_1 + 3c_2 = -1 \tag{1}

Equation 2 (y-component):

2c_2 + c_3 = 1 \tag{2}

Equation 3 (z-component):

c_1 + c_3 = 5 \tag{3}

Step 1: Solve Equation (3) for $c_1$

From Equation (3): c_1 = 5 - c_3 \tag{4}

Step 2: Substitute $c_1$ from Equation (4) into Equation (1)

Substituting $c_1 = 5 - c_3$ into Equation (1): $(5 - c_3) + 3c_2 = -1$ $5 + 3c_2 - c_3 = -1$ 3c_2 - c_3 = -6 \tag{5}

Step 3: Solve the system of Equations (2) and (5)

We now have the system:

1. $2c_2 + c_3 = 1$ (Equation 2)
2. $3c_2 - c_3 = -6$ (Equation 5)

Add the two equations to eliminate $c_3$:

$(2c_2 + c_3) + (3c_2 - c_3) = 1 + (-6)$ $5c_2 = -5$ $c_2 = -1$

Step 4: Find $c_3$

Substitute $c_2 = -1$ into Equation (2): $2(-1) + c_3 = 1$ $-2 + c_3 = 1$ $c_3 = 3$

Step 5: Find $c_1$

Substitute $c_3 = 3$ into Equation (4): $c_1 = 5 - 3 = 2$

Final Answer:

The scalars are: $c_1 = 2, \quad c_2 = -1, \quad c_3 = 3$

Thus, the vector $\mathbf{w} = \langle -1, 1, 5 \rangle$ can be written as:

$\mathbf{w} = 2 \mathbf{v_1} - \mathbf{v_2} + 3 \mathbf{v_3}$

Do you need any further details or clarifications?

Here are some related questions for practice:

1. How can you check if a set of vectors is linearly independent?
2. What is the geometric interpretation of a linear combination of vectors?
3. How do you express a vector as a linear combination in higher dimensions?
4. What conditions make a vector space span?
5. How would the solution change if one of the vectors was linearly dependent on another?

Tip: To solve systems of linear equations efficiently, matrix methods like Gaussian elimination or using matrix inverses can be very helpful.