To determine if the vector b is in the span of the columns of matrix A, we need to solve the following matrix equation:

$A \mathbf{x} = \mathbf{b}$

Where:

$A = \begin{bmatrix} 2 & 8 \\ 7 & 6 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 3 \\ 6 \end{bmatrix}$

This means we need to check if there exists a vector $\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ such that:

$\begin{bmatrix} 2 & 8 \\ 7 & 6 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 3 \\ 6 \end{bmatrix}$

Let's solve this system of linear equations. I'll compute the solution for you.The system of equations has a solution:

$x_1 = 0.6818, \quad x_2 = 0.2045$

Since a solution exists, the vector b is in the span of the columns of the matrix A.

Would you like a more detailed explanation, or do you have any other questions?

Here are 5 related questions to expand on this topic:

1. What does it mean for a vector to be in the span of a matrix's columns?
2. How can the rank of a matrix help in determining if a solution exists?
3. What is the geometric interpretation of the span of vectors?
4. How does the determinant of a matrix relate to the existence of solutions?
5. What is the significance of linear independence in solving systems of equations?

Tip: When solving systems of linear equations, checking the determinant can quickly tell if the matrix is invertible (and whether a unique solution exists).