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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).

Determine if the vector b is in the span of the columns of the matrix A, where A = [[2, 8], [7, 6]] and b = [3, 6].

In this problem, you will learn how to determine if a given vector is in the span of the columns of a matrix. The matrix A is given as [[2, 8], [7, 6]], and the vector b is [3, 6]. The solution involves solving a system of linear equations.

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