**Title: Understanding Change of Basis Matrices****Introduction to Basis Change:**In linear algebra, changing the basis of a vector space involves transforming the vector representations relative to different bases. Here, we focus on finding the change of basis matrix from basis $ B_1 $ to basis $ B_2 $.**Given Bases:**- $ B_1 = \left\{ \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right\} $- $ B_2 = \left\{ \begin{bmatrix} 5 \\ 3 \end{bmatrix}, \begin{bmatrix} 7 \\ 4 \end{bmatrix} \right\} $**Change of Basis Matrix:**The task is to find the change of basis matrix that converts coordinates from $ B_1 $ to $ B_2 $.**Matrix Representation:**The change of basis matrix is represented as:\[\begin{bmatrix}-3 & -1 \\2 & 2 \end{bmatrix}\]- The numbers inside the matrix are arranged in rows and columns. - The elements of the matrix represent linear transformations needed to express vectors in terms of the new basis.**Arrows and Validation:**- Green arrows beside the matrix elements likely indicate a proposed solution to the basis transformation.- A red cross at the bottom left might suggest an incorrect solution or an error in the transformation process, indicating a need to re-evaluate the calculations.**Conclusion:**Clearly understanding and calculating change of basis matrices is essential for manipulating vector representations across different coordinate systems in linear algebra. The matrix above plays a crucial role in transforming vector coordinates from one basis $ B_1 $ to another basis $ B_2 $. **Find the Change of Basis Matrix**To find the change of basis matrix from $ B_2 $ to $ B_1 $, we are given two sets of basis vectors:- $ B_1 = \left\{ \begin{bmatrix} 4 \\ 1 \end{bmatrix}, \begin{bmatrix} 7 \\ 2 \end{bmatrix} \right\} $- $ B_2 = \left\{ \begin{bmatrix} 5 \\ 1 \end{bmatrix}, \begin{bmatrix} 11 \\ 2 \end{bmatrix} \right\} $The image includes a blank 2x2 matrix, which indicates the position for the change of basis matrix.Arrows are included in the image for visual guidance:- A rightward arrow pointing from $ B_2 $ to the blank matrix, suggesting the transformation process.- A downward arrow on the left side of the blank matrix, signifying the order of operations or matrix placement. To solve this problem, you need to express the vectors of $ B_2 $ in terms of the vectors of $ B_1 $ and compute the resulting transformation matrix.

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Answered: Find the change of basis matrix from B1…

Math

Advanced Math

Find the change of basis matrix from B1 to B2. -1 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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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

**Title: Understanding Change of Basis Matrices**

**Introduction to Basis Change:**
In linear algebra, changing the basis of a vector space involves transforming the vector representations relative to different bases. Here, we focus on finding the change of basis matrix from basis $ B_1 $ to basis $ B_2 $.

**Given Bases:**
- $ B_1 = \left\{ \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right\} $
- $ B_2 = \left\{ \begin{bmatrix} 5 \\ 3 \end{bmatrix}, \begin{bmatrix} 7 \\ 4 \end{bmatrix} \right\} $

**Change of Basis Matrix:**
The task is to find the change of basis matrix that converts coordinates from $ B_1 $ to $ B_2 $.

**Matrix Representation:**
The change of basis matrix is represented as:
\[
\begin{bmatrix}
-3 & -1 \\
2 & 2
\end{bmatrix}
\]

- The numbers inside the matrix are arranged in rows and columns.
- The elements of the matrix represent linear transformations needed to express vectors in terms of the new basis.

**Arrows and Validation:**
- Green arrows beside the matrix elements likely indicate a proposed solution to the basis transformation.
- A red cross at the bottom left might suggest an incorrect solution or an error in the transformation process, indicating a need to re-evaluate the calculations.

**Conclusion:**
Clearly understanding and calculating change of basis matrices is essential for manipulating vector representations across different coordinate systems in linear algebra. The matrix above plays a crucial role in transforming vector coordinates from one basis $ B_1 $ to another basis $ B_2 $.

**Find the Change of Basis Matrix**

To find the change of basis matrix from $ B_2 $ to $ B_1 $, we are given two sets of basis vectors:

- $ B_1 = \left\{ \begin{bmatrix} 4 \\ 1 \end{bmatrix}, \begin{bmatrix} 7 \\ 2 \end{bmatrix} \right\} $
- $ B_2 = \left\{ \begin{bmatrix} 5 \\ 1 \end{bmatrix}, \begin{bmatrix} 11 \\ 2 \end{bmatrix} \right\} $

The image includes a blank 2x2 matrix, which indicates the position for the change of basis matrix.

Arrows are included in the image for visual guidance:
- A rightward arrow pointing from $ B_2 $ to the blank matrix, suggesting the transformation process.
- A downward arrow on the left side of the blank matrix, signifying the order of operations or matrix placement.

To solve this problem, you need to express the vectors of $ B_2 $ in terms of the vectors of $ B_1 $ and compute the resulting transformation matrix.

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