show me how to calculate a rotation matrix of a coordinate system between the given start matrix A and the given desired matrix B

To calculate a rotation matrix between two coordinate systems, defined by a **start matrix A** and a **desired matrix B**, the steps below outline the process:

### Steps for Calculating the Rotation Matrix:

1. **Define the Matrices:**
   Let \( A \) and \( B \) represent the initial and desired coordinate systems, respectively. Each matrix should be a 3x3 matrix where each column vector represents the basis vectors of the respective coordinate system.

   \[
   A = \begin{bmatrix} 
   \mathbf{a}_1 & \mathbf{a}_2 & \mathbf{a}_3
   \end{bmatrix}, 
   \quad B = \begin{bmatrix} 
   \mathbf{b}_1 & \mathbf{b}_2 & \mathbf{b}_3
   \end{bmatrix}
   \]
   
   where \( \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \) are the basis vectors of the start matrix \( A \), and \( \mathbf{b}_1, \mathbf{b}_2, \mathbf{b}_3 \) are the basis vectors of the desired matrix \( B \).

2. **Rotation Matrix Concept:**
   The rotation matrix \( R \) that transforms the coordinate system from \( A \) to \( B \) is given by:
   \[
   R = B A^{-1}
   \]
   where \( A^{-1} \) is the inverse of the matrix \( A \).

   The idea is that multiplying by \( A^{-1} \) undoes the rotation in the original system, and multiplying by \( B \) applies the new desired rotation.

3. **Calculate the Inverse of \( A \):**
   To find \( A^{-1} \), the inverse of the start matrix, use standard matrix inversion techniques:
   
   - If \( A \) is orthogonal (i.e., its column vectors are orthonormal), \( A^{-1} = A^\top \), the transpose of \( A \).
   - Otherwise, compute the inverse of \( A \) using Gaussian elimination, adjugate/cofactor method, or direct matrix inversion formulas.

4. **Multiply the Matrices:**
   Once you have \( A^{-1} \), multiply it by the matrix \( B \):
   \[
   R = B A^{-1}
   \]
   Perform standard matrix multiplication to get the rotation matrix \( R \).

5. **Interpret the Result:**
   The resulting matrix \( R \) is the **rotation matrix** that transforms the coordinate system from \( A \) to \( B \). This matrix can be used to rotate vectors from the start system into the desired system.

### Example:

If \( A \) and \( B \) are defined as follows:
\[
A = \begin{bmatrix} 
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}, 
\quad B = \begin{bmatrix} 
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 1
\end{bmatrix}
\]
Here, \( A \) is the identity matrix (standard basis), and \( B \) represents a 90° counterclockwise rotation about the z-axis.

1. The inverse of \( A \) is:
   \[
   A^{-1} = A = \begin{bmatrix} 
   1 & 0 & 0 \\
   0 & 1 & 0 \\
   0 & 0 & 1
   \end{bmatrix

Learn how to calculate a rotation matrix that transforms the coordinate system from matrix A to matrix B. This problem involves key concepts from linear algebra, including matrix inversion and orthogonality. A detailed step-by-step solution is provided for undergraduate-level students.

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