Let B, D be the following two bases of R³: B = D= 0 0 -2 2 a) Find the change of coordinates matrix PDB from the basis B to D. 2 0 -2 }]}
Let B, D be the following two bases of R³: B = D= 0 0 -2 2 a) Find the change of coordinates matrix PDB from the basis B to D. 2 0 -2 }]}
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Let \( \mathcal{B}, \mathcal{D} \) be the following two bases of \( \mathbb{R}^3 \):
\[
\mathcal{B} = \left\{ \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ -2 \\ 0 \end{bmatrix}, \begin{bmatrix} -1 \\ -2 \\ 3 \end{bmatrix} \right\}
\]
\[
\mathcal{D} = \left\{ \begin{bmatrix} -2 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} -1 \\ -4 \\ 0 \end{bmatrix}, \begin{bmatrix} -2 \\ -4 \\ 1 \end{bmatrix} \right\}
\]
a) Find the change of coordinates matrix \( P_{\mathcal{D} \leftarrow \mathcal{B}} \) from the basis \( \mathcal{B} \) to \( \mathcal{D} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb493ecdd-cbdc-400d-a05a-de2010eb2d52%2F467040dd-9045-4700-bb63-8b53d085adb7%2F82s9uq_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( \mathcal{B}, \mathcal{D} \) be the following two bases of \( \mathbb{R}^3 \):
\[
\mathcal{B} = \left\{ \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ -2 \\ 0 \end{bmatrix}, \begin{bmatrix} -1 \\ -2 \\ 3 \end{bmatrix} \right\}
\]
\[
\mathcal{D} = \left\{ \begin{bmatrix} -2 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} -1 \\ -4 \\ 0 \end{bmatrix}, \begin{bmatrix} -2 \\ -4 \\ 1 \end{bmatrix} \right\}
\]
a) Find the change of coordinates matrix \( P_{\mathcal{D} \leftarrow \mathcal{B}} \) from the basis \( \mathcal{B} \) to \( \mathcal{D} \).
![Enter the matrix \( P_{\mathcal{D} \leftarrow \mathcal{B}} \):
[Input box]
b) Let \(\mathbf{v}\) be a vector in \(\mathbb{R}^3\) whose vector of coordinates relative to the basis \(\mathcal{B}\) is given by
\[
[\mathbf{v}]_{\mathcal{B}} = \begin{bmatrix} -1 \\ 4 \\ -1 \end{bmatrix}
\]
Find the vector \([\mathbf{v}]_{\mathcal{D}}\) of coordinates of \(\mathbf{v}\) relative to the basis of \(\mathcal{D}\).
Enter the vector \([\mathbf{v}]_{\mathcal{D}}\) in the form \([c_1, c_2, c_3]\):
[Input box]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb493ecdd-cbdc-400d-a05a-de2010eb2d52%2F467040dd-9045-4700-bb63-8b53d085adb7%2Fsd5luio_processed.png&w=3840&q=75)
Transcribed Image Text:Enter the matrix \( P_{\mathcal{D} \leftarrow \mathcal{B}} \):
[Input box]
b) Let \(\mathbf{v}\) be a vector in \(\mathbb{R}^3\) whose vector of coordinates relative to the basis \(\mathcal{B}\) is given by
\[
[\mathbf{v}]_{\mathcal{B}} = \begin{bmatrix} -1 \\ 4 \\ -1 \end{bmatrix}
\]
Find the vector \([\mathbf{v}]_{\mathcal{D}}\) of coordinates of \(\mathbf{v}\) relative to the basis of \(\mathcal{D}\).
Enter the vector \([\mathbf{v}]_{\mathcal{D}}\) in the form \([c_1, c_2, c_3]\):
[Input box]
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