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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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Just answer 3c
![**Problem 3:**
For each of the following linear operators \( T \) on a vector space \( V \) and ordered bases \( \beta \), compute \([T]_{\beta}\), and determine whether \( \beta \) is a basis consisting of eigenvectors of \( T \).
---
**(a)**
- \( V = \mathbb{R}^2 \)
- \( T \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} 10a - 6b \\ 17a - 10b \end{pmatrix} \)
- \( \beta = \left\{ \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 3 \end{pmatrix} \right\} \)
---
**(b)**
- \( V = P_1(\mathbb{R}) \)
- \( T(a + bx) = (6a - 6b) + (12a - 11b)x \)
- \( \beta = \{3 + 4x, 2 + 3x\} \)
---
**(c)**
- \( V = \mathbb{R}^3 \)
- \( T \begin{pmatrix} a \\ b \\ c \end{pmatrix} = \begin{pmatrix} 3a + 2b - 2c \\ -4a - 3b + 2c \\ -c \end{pmatrix} \)
- \( \beta = \left\{ \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} \right\} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2F7a3e167c-a72d-4b26-9ac3-159cae00287b%2F277d6qp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3:**
For each of the following linear operators \( T \) on a vector space \( V \) and ordered bases \( \beta \), compute \([T]_{\beta}\), and determine whether \( \beta \) is a basis consisting of eigenvectors of \( T \).
---
**(a)**
- \( V = \mathbb{R}^2 \)
- \( T \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} 10a - 6b \\ 17a - 10b \end{pmatrix} \)
- \( \beta = \left\{ \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 3 \end{pmatrix} \right\} \)
---
**(b)**
- \( V = P_1(\mathbb{R}) \)
- \( T(a + bx) = (6a - 6b) + (12a - 11b)x \)
- \( \beta = \{3 + 4x, 2 + 3x\} \)
---
**(c)**
- \( V = \mathbb{R}^3 \)
- \( T \begin{pmatrix} a \\ b \\ c \end{pmatrix} = \begin{pmatrix} 3a + 2b - 2c \\ -4a - 3b + 2c \\ -c \end{pmatrix} \)
- \( \beta = \left\{ \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} \right\} \)
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