**Change of Basis Matrix Calculation**To find the change of basis matrix from basis $ B_2 $ to $ B_1 $, we need to perform transformations using the given matrices.**Given Bases:**- **Basis $ B_1 $:** \[ B_1 = \left\{ \begin{bmatrix} -1 \\ 4 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} \right\} \]- **Basis $ B_2 $:** \[ B_2 = \left\{ \begin{bmatrix} 1 \\ 7 \\ 3 \end{bmatrix}, \begin{bmatrix} -4 \\ 5 \\ 3 \end{bmatrix} \right\} \]**Steps to Find the Change of Basis Matrix:**1. Construct the transformation matrix using the vectors from both bases.2. Use these matrices to find the linear transformation that maps vectors from the second basis $ B_2 $ to the first basis $ B_1 $.**Diagram Explanation:**- The diagram includes a 2x2 placeholder matrix with arrows pointing to and from it.- This represents the change of basis matrix that you will compute.- The arrows indicate the direction of transformation from one basis to another.This matrix is essential in applications such as computer graphics, where different coordinate systems are used, or in solving linear equations where different reference frames are considered. Understanding how to switch between bases efficiently is crucial for computations involving linear transformations.

The change of basis matrix C from B2 to B1( or from the basis B2 to the basis B1 ): β1=-142,231 and…

Answered: Find the change of basis matrix from B2…

Math

Advanced Math

Find the change of basis matrix from B2 to B1. -4 B1 = , B2 B2 = 2

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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**Change of Basis Matrix Calculation**

To find the change of basis matrix from basis $ B_2 $ to $ B_1 $, we need to perform transformations using the given matrices.

**Given Bases:**

- **Basis $ B_1 $:**
\[
B_1 = \left\{ \begin{bmatrix} -1 \\ 4 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} \right\}
\]

- **Basis $ B_2 $:**
\[
B_2 = \left\{ \begin{bmatrix} 1 \\ 7 \\ 3 \end{bmatrix}, \begin{bmatrix} -4 \\ 5 \\ 3 \end{bmatrix} \right\}
\]

**Steps to Find the Change of Basis Matrix:**

1. Construct the transformation matrix using the vectors from both bases.
2. Use these matrices to find the linear transformation that maps vectors from the second basis $ B_2 $ to the first basis $ B_1 $.

**Diagram Explanation:**

- The diagram includes a 2x2 placeholder matrix with arrows pointing to and from it.
- This represents the change of basis matrix that you will compute.
- The arrows indicate the direction of transformation from one basis to another.

This matrix is essential in applications such as computer graphics, where different coordinate systems are used, or in solving linear equations where different reference frames are considered. Understanding how to switch between bases efficiently is crucial for computations involving linear transformations.

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