https://file.aiquickdraw.com/mathbot/file/1726086295876l9pevmpe.jpg

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.

Write down the matrices that take [x]_B to [x]_C. Let B = {[3 -5], [4 -6]} and C = {[4 5], [6 -7]}.

This problem involves finding the matrix that converts coordinates of a vector from basis B = {[3 -5], [4 -6]} to basis C = {[4 5], [6 -7]}. The solution demonstrates key concepts in Linear Algebra including matrix transformations and the change of basis formula.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation