The lab task in the image involves performing a linear combination of two vectors:

$\mathbf{v} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad \mathbf{w} = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$

The goal is to find scalars $a$ and $b$ such that the linear combination:

$a\mathbf{v} + b\mathbf{w} = \begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}$

This is represented by the equation:

$a \begin{bmatrix} 1 \\ 1 \end{bmatrix} + b \begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}$

This can be broken down into a system of equations by multiplying and adding the vectors component-wise:

$a + 3b = 9.4$ $a + b = 6.4$

Now, let's solve this system of equations to find the values of $a$ and $b$. I'll proceed with the calculations.The solution to the system of equations is:

$a = 4.9, \quad b = 1.5$

Thus, the scalars $a = 4.9$ and $b = 1.5$ will give a linear combination of the vectors $\mathbf{v}$ and $\mathbf{w}$ that results in the vector $\begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}$.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How can you geometrically interpret the linear combination of two vectors?
2. What happens when one of the scalars $a$ or $b$ is zero in the linear combination?
3. How would you solve this system of equations if the resulting vector were different?
4. Can the vectors $\mathbf{v}$ and $\mathbf{w}$ span the entire 2D plane? Why or why not?
5. How can you generalize this method to higher dimensions with more vectors?

Tip: Solving systems of linear equations is fundamental for finding relationships between vectors. Practice solving such systems using substitution and elimination methods.