The problem in your image is asking to create a linear combination of two vectors $\vec{v} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ and $\vec{w} = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$, by adjusting the scalars $a$ and $b$, so that the result matches the vector $\begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}$. The combination of the two vectors can be written as:

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

Substituting the components of $\vec{v}$ and $\vec{w}$ into the equation, we get:

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

This leads to the following system of equations:

1. $a + 3b = 9.4$
2. $a + b = 6.4$

We can solve this system of equations to find the values of $a$ and $b$. Let me calculate them for you.The solution to the system of equations is:

• $a = 4.9$
• $b = 1.5$

Therefore, the linear combination that matches the vector $\begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}$ is:

$4.9 \cdot \vec{v} + 1.5 \cdot \vec{w}$

Would you like a more detailed explanation or have any further questions?

Here are five related questions to expand your understanding:

1. How can you geometrically interpret the linear combination of two vectors?
2. What happens if you choose different values for $a$ and $b$?
3. How do linear combinations help in determining vector spaces?
4. Can you visualize what it means for a set of vectors to span a space?
5. How would the solution change if the target vector were different?

Tip: Linear combinations of vectors form the foundation for many applications, such as solving systems of equations and defining vector spaces.