386 por transformation of vector sp (b) Find [T]B,B.
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 help me solve just part be please?
![**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
\[
B = \{ \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \} = \left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} \right\}
\]
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%2F52fb4760-8b56-41d1-811f-26cd2f9d5b41%2Fc81d6483-dbea-472c-8831-5115ae5f2160%2Fs2ang1_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
\[
B = \{ \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \} = \left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} \right\}
\]
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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