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The problem is asking to find the change-of-coordinates matrix from basis \( C \) to basis \( B \) for \( \mathbb{R}^2 \).

We are given two bases:

- \( B = \left\{ \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \begin{pmatrix} 3 \\ 1 \end{pmatrix} \right\} \)
- \( C = \left\{ \begin{pmatrix} 4 \\ 0 \end{pmatrix}, \begin{pmatrix} 18 \\ 7 \end{pmatrix} \right\} \)

### Steps to find the change-of-coordinates matrix:
1. **Express the vectors in \( C \) in terms of the vectors in \( B \).**
   - We need to solve for scalars \( a_1, a_2 \) such that:
     \[
     \begin{pmatrix} 4 \\ 0 \end{pmatrix} = a_1 \begin{pmatrix} 2 \\ 1 \end{pmatrix} + a_2 \begin{pmatrix} 3 \\ 1 \end{pmatrix}
     \]
     - This gives a system of linear equations.
   
   Similarly, find \( b_1, b_2 \) such that:
     \[
     \begin{pmatrix} 18 \\ 7 \end{pmatrix} = b_1 \begin{pmatrix} 2 \\ 1 \end{pmatrix} + b_2 \begin{pmatrix} 3 \\ 1 \end{pmatrix}
     \]

2. **Solve the systems of equations.**
   These will give the coordinates of the vectors in \( C \) in terms of the basis \( B \).

3. **Form the change-of-coordinates matrix.**
   The matrix is formed by placing the coordinates of each vector from \( C \) in terms of \( B \) as columns of the matrix.

I'll now calculate these steps for you:

#### For the first vector in \( C \), \( \begin{pmatrix} 4 \\ 0 \end{pmatrix} \):
\[
a_1 \begin{pmatrix} 2 \\ 1 \end{pmatrix} + a_2 \begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} 4 \\ 0 \end{pmatrix}
\]
- First equation (from x-components): \( 2a_1 + 3a_2 = 4 \)
- Second equation (from y-components): \( a_1 + a_2 = 0 \)

From the second equation, we have \( a_1 = -a_2 \).

Substitute into the first equation:
\[
2(-a_2) + 3a_2 = 4 \quad \Rightarrow \quad -2a_2 + 3a_2 = 4 \quad \Rightarrow \quad a_2 = 4
\]
Therefore, \( a_1 = -4 \).

So, \( \begin{pmatrix} 4 \\ 0 \end{pmatrix} \) in terms of \( B \) is \( \begin{pmatrix} -4 \\ 4 \end{pmatrix} \).

#### For the second vector in \( C \), \( \begin{pmatrix} 18 \\ 7 \end{pmatrix} \):
\[
b_1 \begin{pmatrix} 2 \\ 1 \end{pmatrix} + b_2 \begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} 18 \\ 7 \end{pmatrix}
\]
- First equation: \( 2b_1 + 3b_2 = 18 \)
- Second equation: \( b_1 + b_2 = 7 \)

From the second equation, \( b_1 = 7 - b_2 \).

Substitute into the first equation:
\[
2(7 - b_2) + 3b_2 = 18 \quad \Rightarrow \quad 14 - 2b_2 + 3b_2 = 18 \quad \Rightarrow \quad b_2 = 4
\]
Thus, \( b_1 = 3 \).

So, \( \begin{pmatrix} 18 \\ 7 \end{pmatrix} \) in terms of \( B \) is \( \begin{pmatrix} 3 \\ 4 \end{pmatrix} \).

### Final Change-of-Coordinates Matrix:
The change-of-coordinates matrix from \( C \) to \( B \) is given by:
\[
P_{C \to B} = \begin{pmatrix} -4 & 3 \\ 4 & 4 \end{pmatrix}
\]

Would you like a detailed explanation of any part of the process or have any specific questions?

Here are 5 related questions to think about:
1. How do we verify if two sets of vectors form a basis for \( \mathbb{R}^2 \)?
2. What is the inverse of the change-of-coordinates matrix \( P_{C \to B} \)?
3. How would this process change if we were working in \( \mathbb{R}^3 \) instead of \( \mathbb{R}^2 \)?
4. How do we apply the change-of-coordinates matrix to convert a vector from one basis to another?
5. What properties does the matrix \( P_{C \to B} \) need to satisfy to be invertible?

**Tip:** When solving systems of linear equations, always check if the equations are independent to ensure a unique solution exists!

Find the change-of-coordinates matrix from basis C to basis B where: 
B = { [2 1], [3 1] } and C = { [4 0], [18 7] } are bases for R^2.

Learn how to find the change-of-coordinates matrix from basis C to basis B in R^2. This problem involves solving systems of linear equations and applying key linear algebra concepts like matrix multiplication and basis transformation.

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