The problem asks us to find the matrix that takes the coordinates of a vector $x$ with respect to basis $\mathcal{B}$ to the coordinates of $x$ with respect to basis $\mathcal{C}$.

Given Information:

• The basis $\mathcal{B}$ consists of the vectors $\left\{ \begin{bmatrix} 3 \\ -5 \end{bmatrix}, \begin{bmatrix} 4 \\ -6 \end{bmatrix} \right\}$.
• The basis $\mathcal{C}$ consists of the vectors $\left\{ \begin{bmatrix} 4 \\ 5 \end{bmatrix}, \begin{bmatrix} 6 \\ -7 \end{bmatrix} \right\}$.

Goal:

We need to find the matrix that transforms a vector's coordinates from basis $\mathcal{B}$ to basis $\mathcal{C}$. This involves finding the change-of-basis matrix from $\mathcal{B}$ to $\mathcal{C}$.

Procedure:

1. Find the matrix representation of the vectors in $\mathcal{B}$:
   $\mathcal{B} = \left\{ \begin{bmatrix} 3 \\ -5 \end{bmatrix}, \begin{bmatrix} 4 \\ -6 \end{bmatrix} \right\}$.

2. Find the matrix representation of the vectors in $\mathcal{C}$:
   $\mathcal{C} = \left\{ \begin{bmatrix} 4 \\ 5 \end{bmatrix}, \begin{bmatrix} 6 \\ -7 \end{bmatrix} \right\}$.

3. Express the basis vectors of $\mathcal{B}$ in terms of the vectors of $\mathcal{C}$:
   To do this, we need to solve for the coordinates of each vector in $\mathcal{B}$ in terms of $\mathcal{C}$'s basis vectors. Let the change of basis matrix from $\mathcal{B}$ to $\mathcal{C}$ be $P_{\mathcal{B} \to \mathcal{C}}$. We can set up the system of equations for each vector in $\mathcal{B}$.

I will compute this matrix transformation now.The matrix that transforms the coordinates of a vector from basis $\mathcal{B}$ to basis $\mathcal{C}$ is:

$P_{\mathcal{B} \to \mathcal{C}} = \begin{bmatrix} -22 & -4 \\ 17.5 & 4.5 \end{bmatrix}$

This matrix can be used to convert any vector's coordinates in the $\mathcal{B}$-basis to its coordinates in the $\mathcal{C}$-basis.

Would you like further details or explanations?

5 Related Questions:

1. How do you verify a change of basis matrix?
2. What are the properties of invertible matrices?
3. Can a vector have more than one coordinate representation depending on the basis?
4. What is the geometric interpretation of a change of basis?
5. How do you find the inverse of a matrix?

Tip:

To switch between different bases efficiently, always ensure that both bases are linearly independent. Otherwise, the transformation matrix will not exist.