To find the $B$-coordinate vector of $\mathbf{x}$ relative to the basis $B = \{ \mathbf{b}_1, \mathbf{b}_2 \}$, we need to express $\mathbf{x}$ as a linear combination of $\mathbf{b}_1$ and $\mathbf{b}_2$.

That is, we want to find scalars $c_1$ and $c_2$ such that:

$\mathbf{x} = c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2$

Step 1: Set up the system of equations

The vectors $\mathbf{b}_1$, $\mathbf{b}_2$, and $\mathbf{x}$ are given as:

$\mathbf{b}_1 = \begin{bmatrix} 4 \\ -15 \end{bmatrix}, \quad \mathbf{b}_2 = \begin{bmatrix} -1 \\ 5 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} 4 \\ -10 \end{bmatrix}$

We now write the equation:

$\begin{bmatrix} 4 \\ -10 \end{bmatrix} = c_1 \begin{bmatrix} 4 \\ -15 \end{bmatrix} + c_2 \begin{bmatrix} -1 \\ 5 \end{bmatrix}$

This translates to the system of linear equations:

$4 = 4c_1 - c_2$ $-10 = -15c_1 + 5c_2$

Step 2: Solve the system of equations

We solve this system step by step.

1. From the first equation, solve for $c_2$:

$c_2 = 4c_1 - 4$

2. Substitute this expression for $c_2$ into the second equation:

$-10 = -15c_1 + 5(4c_1 - 4)$ $-10 = -15c_1 + 20c_1 - 20$ $-10 + 20 = 5c_1$ $10 = 5c_1$ $c_1 = 2$

3. Substitute $c_1 = 2$ back into the equation for $c_2$:

$c_2 = 4(2) - 4 = 8 - 4 = 4$

Step 3: Write the $B$-coordinate vector

The $B$-coordinate vector of $\mathbf{x}$ is:

$\left[ \mathbf{x} \right]_B = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \end{bmatrix}$

Final Answer:

$\left[ \mathbf{x} \right]_B = \begin{bmatrix} 2 \\ 4 \end{bmatrix}$

Would you like more details on any step or have further questions? Here are some related questions you can explore:

1. How do you interpret the $B$-coordinate vector geometrically?
2. What is the significance of finding coordinates relative to a given basis?
3. How do you check the solution by reconstructing $\mathbf{x}$?
4. What happens if the vectors $\mathbf{b}_1$ and $\mathbf{b}_2$ are not linearly independent?
5. How do you calculate the $B$-coordinate vector for a higher-dimensional space?

Tip: Always double-check the steps when solving linear systems to avoid arithmetic errors!