**Linear Algebra Problem: Diagonalizing a Matrix****Introduction:**In this exercise, we are given a linear transformation $ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ defined by $ T(x) = Ax $, where $ A $ is a specific matrix. Our goal is to find a basis $ B $ for $ \mathbb{R}^2 $ such that the matrix representation of $ T $ in this basis, denoted by $[T]_B$, is diagonal.**Given:**The matrix $ A $ is defined as follows:\[ A = \begin{pmatrix} -4 & -6 \\ 9 & 11 \end{pmatrix} \]**Task:**Find a basis $ B $ for $ \mathbb{R}^2 $ such that $[T]_B$ is diagonal.**Solution Steps:**1. **Matrix A:** The given matrix $ A $ is: \[ A = \begin{pmatrix} -4 & -6 \\ 9 & 11 \end{pmatrix} \]2. **Basis $ B $:** We are aiming to find a basis $ B $ such that: \[ B = \left\{ \begin{pmatrix} 1 \\ -1 \end{pmatrix}, \begin{pmatrix} \boxed{} \\ \boxed{} \end{pmatrix} \right\} \] Here, the second basis vector needs to be determined and is currently left blank.3. **Transformation $ [T]_B $:** To find the matrix representation of $ T $ in the basis $ B $ which is diagonal: \[ [T]_B = \begin{pmatrix} \boxed{} & \boxed{} \\ \boxed{} & \boxed{} \end{pmatrix} \] The elements of this diagonal matrix need to be determined.**Conclusion:**To finalize the solution, we need to calculate the eigenvalues of the matrix $ A $ and find the corresponding eigenvectors which will form the basis $ B $ in which the transformation matrix $[T]_B$ is diagonal.By

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Answered: Define T: R-R by Tix) =Ax, where Ais…

Define T: R-R by Tix) =Ax, where Ais the matrix defined below. Find a basis B for R? with the property that (Te is diagonal.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

**Linear Algebra Problem: Diagonalizing a Matrix**

**Introduction:**
In this exercise, we are given a linear transformation $ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ defined by $ T(x) = Ax $, where $ A $ is a specific matrix. Our goal is to find a basis $ B $ for $ \mathbb{R}^2 $ such that the matrix representation of $ T $ in this basis, denoted by $[T]_B$, is diagonal.

**Given:**
The matrix $ A $ is defined as follows:

\[
A = \begin{pmatrix}
-4 & -6 \\
9 & 11
\end{pmatrix}
\]

**Task:**
Find a basis $ B $ for $ \mathbb{R}^2 $ such that $[T]_B$ is diagonal.

**Solution Steps:**

1. **Matrix A:**
The given matrix $ A $ is:
\[
A = \begin{pmatrix}
-4 & -6 \\
9 & 11
\end{pmatrix}
\]

2. **Basis $ B $:**
We are aiming to find a basis $ B $ such that:
\[
B = \left\{ \begin{pmatrix} 1 \\ -1 \end{pmatrix}, \begin{pmatrix} \boxed{} \\ \boxed{} \end{pmatrix} \right\}
\]
Here, the second basis vector needs to be determined and is currently left blank.

3. **Transformation $ [T]_B $:**
To find the matrix representation of $ T $ in the basis $ B $ which is diagonal:
\[
[T]_B = \begin{pmatrix}
\boxed{} & \boxed{} \\
\boxed{} & \boxed{}
\end{pmatrix}
\]
The elements of this diagonal matrix need to be determined.

**Conclusion:**
To finalize the solution, we need to calculate the eigenvalues of the matrix $ A $ and find the corresponding eigenvectors which will form the basis $ B $ in which the transformation matrix $[T]_B$ is diagonal.

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