a. Find the transition matrix from the ordered basis B1 to the ordered basis B2. b. Find the coordinates of the vector
a. Find the transition matrix from the ordered basis B1 to the ordered basis B2. b. Find the coordinates of the vector
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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Question
![**Problem Statement:**
21. Let
\[
B_1 = \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\}
\]
be the standard ordered basis for \(\mathbb{R}^3\) and let
\[
B_2 = \left\{ \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\}
\]
be a second ordered basis.
a. Find the transition matrix from the ordered basis \(B_1\) to the ordered basis \(B_2\).
b. Find the coordinates of the vector
\[
\mathbf{v} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}
\]
relative to the ordered basis \(B_2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa00d0151-b5df-4ee3-a0d6-fdfba7d1c4f7%2Fec4cb9d2-e9aa-482e-a7f9-2544ad40e8bb%2F50mpsbn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
21. Let
\[
B_1 = \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\}
\]
be the standard ordered basis for \(\mathbb{R}^3\) and let
\[
B_2 = \left\{ \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\}
\]
be a second ordered basis.
a. Find the transition matrix from the ordered basis \(B_1\) to the ordered basis \(B_2\).
b. Find the coordinates of the vector
\[
\mathbf{v} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}
\]
relative to the ordered basis \(B_2\).
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