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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.

The lab task involves performing a linear combination of two vectors: v = [1, 1] and w = [3, 1]. The goal is to find scalars a and b such that a * v + b * w results in the vector [9.4, 6.4].

Explore how to find scalars a and b in the linear combination of vectors v = [1, 1] and w = [3, 1], resulting in the vector [9.4, 6.4]. This problem covers vector algebra and solving systems of linear equations. Suitable for students in Grades 10-12 and early college levels.

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