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!