The problem involves finding scalars $a$ and $b$ such that the linear combination of the given vectors $\mathbf{v} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ and $\mathbf{w} = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$ equals the target vector $\begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}$.

Step-by-Step Solution:

1. Define the linear combination: The linear combination of vectors $\mathbf{v}$ and $\mathbf{w}$ is given by: $a\mathbf{v} + b\mathbf{w} = a\begin{bmatrix} 1 \\ 1 \end{bmatrix} + b\begin{bmatrix} 3 \\ 1 \end{bmatrix}.$

2. Expand the combination: This can be rewritten as: $\begin{bmatrix} a + 3b \\ a + b \end{bmatrix}.$

3. Set up the system of equations: We need this to equal the target vector: $\begin{bmatrix} a + 3b \\ a + b \end{bmatrix} = \begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}.$ This gives us two equations: $a + 3b = 9.4,$ $a + b = 6.4.$

4. Solve the system: Subtract the second equation from the first: $(a + 3b) - (a + b) = 9.4 - 6.4,$ $2b = 3,$ $b = 1.5.$

   Substitute $b = 1.5$ into the second equation: $a + 1.5 = 6.4,$ $a = 4.9.$

Final Answer:

The scalars are $a = 4.9$ and $b = 1.5$.

Would you like more details or have any questions?

Here are some related questions you might find helpful:

1. What is the geometric interpretation of linear combinations?
2. How do different values of $a$ and $b$ affect the resulting vector?
3. Can vectors $\mathbf{v}$ and $\mathbf{w}$ span all of $\mathbb{R}^2$?
4. What is the significance of solving systems of linear equations in vector spaces?
5. How can you determine if two vectors are linearly independent?

Tip: Understanding the geometric representation of vectors helps in visualizing the effect of different scalar values on vector combinations.