Suppose we have the following two bases for R²:
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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![The image contains a problem about changing bases in \(\mathbb{R}^2\).
---
**II. Suppose we have the following two bases for \(\mathbb{R}^2\):**
\[
\mathcal{B} = \left\langle \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right\rangle \quad \text{and} \quad \mathcal{D} = \left\langle \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \end{bmatrix} \right\rangle
\]
Let \(\vec{v} = \begin{bmatrix} 4 \\ -1 \end{bmatrix}\).
- **Find** \( \text{Rep}_{\mathcal{B}}(\vec{v}) \) **and** \( \text{Rep}_{\mathcal{D}}(\vec{v}) \).
\[
\text{Rep}_{\mathcal{B}}(\vec{v}) = \quad \text{Rep}_{\mathcal{D}}(\vec{v}) =
\]
- **Find** \( \text{Rep}_{\mathcal{B,D}}(id) \), **the Change of Basis matrix from** \(\mathbb{R}^2_{\mathcal{B}}\) **to** \(\mathbb{R}^2_{\mathcal{D}}\).
- **Use that Change of Basis matrix to find** \( \text{Rep}_{\mathcal{D}}(\vec{v}) \).
Compare your answer with the one you obtained at the beginning of the problem.
\[
\text{Rep}_{\mathcal{D}}(\vec{v}) =
\]
---
**Now, consider the standard basis \(\varepsilon_2\) for \(\mathbb{R}^2\).**
- **Let** \(\mathcal{B} = \left\langle \text{Rep}_{\varepsilon_2}\left(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\right), \text{Rep}_{\varepsilon_2}\left(\begin{b](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa9a9ea7-0a60-40f2-88c8-00f846edd995%2F583346c6-1f37-4938-8e1a-a3c18b7fce59%2Fab3zqy_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains a problem about changing bases in \(\mathbb{R}^2\).
---
**II. Suppose we have the following two bases for \(\mathbb{R}^2\):**
\[
\mathcal{B} = \left\langle \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right\rangle \quad \text{and} \quad \mathcal{D} = \left\langle \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \end{bmatrix} \right\rangle
\]
Let \(\vec{v} = \begin{bmatrix} 4 \\ -1 \end{bmatrix}\).
- **Find** \( \text{Rep}_{\mathcal{B}}(\vec{v}) \) **and** \( \text{Rep}_{\mathcal{D}}(\vec{v}) \).
\[
\text{Rep}_{\mathcal{B}}(\vec{v}) = \quad \text{Rep}_{\mathcal{D}}(\vec{v}) =
\]
- **Find** \( \text{Rep}_{\mathcal{B,D}}(id) \), **the Change of Basis matrix from** \(\mathbb{R}^2_{\mathcal{B}}\) **to** \(\mathbb{R}^2_{\mathcal{D}}\).
- **Use that Change of Basis matrix to find** \( \text{Rep}_{\mathcal{D}}(\vec{v}) \).
Compare your answer with the one you obtained at the beginning of the problem.
\[
\text{Rep}_{\mathcal{D}}(\vec{v}) =
\]
---
**Now, consider the standard basis \(\varepsilon_2\) for \(\mathbb{R}^2\).**
- **Let** \(\mathcal{B} = \left\langle \text{Rep}_{\varepsilon_2}\left(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\right), \text{Rep}_{\varepsilon_2}\left(\begin{b
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