Exercise 10.4.3 Suppose T: R³ R³ is a linear transformation such that →>> be another basis. 1 -8-8-8-8-8-8 T T (b) Find [TB,B 3 Let E= {e1,e2, e3} be the standard basis of R³, and let 3 B = {V1, V2, V3} - (a) Find the matrix of T with respect to E, i.e., find [T]E,E- T 0 {B-Q-A} = 3
Exercise 10.4.3 Suppose T: R³ R³ is a linear transformation such that →>> be another basis. 1 -8-8-8-8-8-8 T T (b) Find [TB,B 3 Let E= {e1,e2, e3} be the standard basis of R³, and let 3 B = {V1, V2, V3} - (a) Find the matrix of T with respect to E, i.e., find [T]E,E- T 0 {B-Q-A} = 3
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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can you please solve (a ) and ( b ) for this question?
![**Exercise 10.4.3**
Suppose \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) is a linear transformation such that
\[ T \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 3 \end{bmatrix}, \quad T \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \quad T \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \\ -1 \end{bmatrix}. \]
Let \( E = \{ \mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3} \} \) be the standard basis of \( \mathbb{R}^3 \), and let
\[ B = \{ \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \} = \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} \right\} \]
be another basis.
(a) Find the matrix of \( T \) with respect to \( E \), i.e., find \( [T]_{E,E} \).
(b) Find \( [T]_{B,B} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98a85d3f-1221-4cd9-b5c2-c225eed9a46f%2Fad9be436-11fd-48c6-8002-2a29d50f7a4c%2Fgy86c3m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 10.4.3**
Suppose \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) is a linear transformation such that
\[ T \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 3 \end{bmatrix}, \quad T \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \quad T \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \\ -1 \end{bmatrix}. \]
Let \( E = \{ \mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3} \} \) be the standard basis of \( \mathbb{R}^3 \), and let
\[ B = \{ \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \} = \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} \right\} \]
be another basis.
(a) Find the matrix of \( T \) with respect to \( E \), i.e., find \( [T]_{E,E} \).
(b) Find \( [T]_{B,B} \).
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