The problem asks whether the vector b is in the span of the vectors a₁ and a₂. We need to determine if there exist scalars $x_1$ and $x_2$ such that:

$b = x_1 a_1 + x_2 a_2$

This translates to solving the following vector equation:

$\begin{pmatrix} -42 \\ -16 \\ -4 \end{pmatrix} = x_1 \begin{pmatrix} -9 \\ -8 \\ 4 \end{pmatrix} + x_2 \begin{pmatrix} -1 \\ -8 \\ 8 \end{pmatrix}$

To solve this, we can set up the corresponding system of linear equations:

1. $-42 = -9x_1 - x_2$
2. $-16 = -8x_1 - 8x_2$
3. $-4 = 4x_1 + 8x_2$

Let me solve this system to check if a solution exists.The solution to the system of equations is $x_1 = 5$ and $x_2 = -3$. This means that the vector b is in the span of the vectors a₁ and a₂, and we can express:

$b = 5a_1 - 3a_2$

Thus, the correct answer is:

• A. Yes, b is in the span.

The coefficients are:

• $x_1 = 5$
• $x_2 = -3$

Would you like a further explanation on how to solve the system of equations?

Here are some related questions to expand your understanding:

1. How do you determine if vectors are linearly independent?
2. What does it mean for a set of vectors to span a vector space?
3. Can vectors spanning a space form a basis for that space?
4. How can matrix methods (e.g., row reduction) be used to check if vectors are in a span?
5. What is the geometric interpretation of a vector being in the span of two others?

Tip: When solving systems of equations, ensure you align the corresponding components of each vector to form the correct system of linear equations.