We are working in R². Let & be the standard basis, B be the basis formed by {(2, 1), (0, −1)}, and C the basis formed by {(1, 1), (-1, 1)}. These vectors and their spans can be visualized below in 'graph paper' form: □ 1. Find the vector that reaches the point Q from the origin in the notation of each basis, that is, find [], [TB, and []c. Q=[³] Q[²] [³] = [²]. E [*] 2. Convert B to B-coordinates and also to C-coordinates. L-0.5_ C
We are working in R². Let & be the standard basis, B be the basis formed by {(2, 1), (0, −1)}, and C the basis formed by {(1, 1), (-1, 1)}. These vectors and their spans can be visualized below in 'graph paper' form: □ 1. Find the vector that reaches the point Q from the origin in the notation of each basis, that is, find [], [TB, and []c. Q=[³] Q[²] [³] = [²]. E [*] 2. Convert B to B-coordinates and also to C-coordinates. L-0.5_ C
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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Hello there, can you please solve a problem? I solved the first two problems but I'm struggling to solve the last problem? Thank you!
![We are working in \(\mathbb{R}^2\). Let \(\mathcal{E}\) be the standard basis, \(\mathcal{B}\) be the basis formed by \(\{(2, 1), (0, -1)\}\), and \(\mathcal{C}\) the basis formed by \(\{(1, 1), (-1, 1)\}\). These vectors and their spans can be visualized below in ‘graph paper’ form:
[Graph images]
- **Left Chart (\(\mathcal{E}\))**: Shows the standard unit vectors with point \(Q\).
- **Middle Chart (\(\mathcal{B}\))**: Shows transformation with basis \((2, 1), (0, -1)\).
- **Right Chart (\(\mathcal{C}\))**: Shows transformation with basis \((1, 1), (-1, 1)\).
1. **Find the vector \(\vec{x}\) that reaches the point \(Q\) from the origin in the notation of each basis**, that is, find \([\vec{x}]_{\mathcal{E}}\), \([\vec{x}]_{\mathcal{B}}\), and \([\vec{x}]_{\mathcal{C}}\).
\[
Q = \begin{bmatrix} 2 \\ 2 \end{bmatrix}_{\mathcal{E}}, \quad
Q = \begin{bmatrix} 1 \\ 0 \end{bmatrix}_{\mathcal{B}}, \quad
Q = \begin{bmatrix} 2 \\ 0 \end{bmatrix}_{\mathcal{C}}
\]
2. **Convert** \(\begin{bmatrix} 2 \\ -3 \end{bmatrix}_{\mathcal{E}}\) **to** \(\mathcal{B}\)-**coordinates and also to** \(\mathcal{C}\)-**coordinates**.
\[
\begin{bmatrix} 1 \\ 4 \end{bmatrix}_{\mathcal{B}}, \quad
\begin{bmatrix} -2.5 \\ -0.5 \end{bmatrix}_{\mathcal{C}}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe00cebfb-aec4-474d-8979-79a2d105b819%2F27ec84d6-d014-4b5c-ae33-3383f702b5c4%2F7etrhd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:We are working in \(\mathbb{R}^2\). Let \(\mathcal{E}\) be the standard basis, \(\mathcal{B}\) be the basis formed by \(\{(2, 1), (0, -1)\}\), and \(\mathcal{C}\) the basis formed by \(\{(1, 1), (-1, 1)\}\). These vectors and their spans can be visualized below in ‘graph paper’ form:
[Graph images]
- **Left Chart (\(\mathcal{E}\))**: Shows the standard unit vectors with point \(Q\).
- **Middle Chart (\(\mathcal{B}\))**: Shows transformation with basis \((2, 1), (0, -1)\).
- **Right Chart (\(\mathcal{C}\))**: Shows transformation with basis \((1, 1), (-1, 1)\).
1. **Find the vector \(\vec{x}\) that reaches the point \(Q\) from the origin in the notation of each basis**, that is, find \([\vec{x}]_{\mathcal{E}}\), \([\vec{x}]_{\mathcal{B}}\), and \([\vec{x}]_{\mathcal{C}}\).
\[
Q = \begin{bmatrix} 2 \\ 2 \end{bmatrix}_{\mathcal{E}}, \quad
Q = \begin{bmatrix} 1 \\ 0 \end{bmatrix}_{\mathcal{B}}, \quad
Q = \begin{bmatrix} 2 \\ 0 \end{bmatrix}_{\mathcal{C}}
\]
2. **Convert** \(\begin{bmatrix} 2 \\ -3 \end{bmatrix}_{\mathcal{E}}\) **to** \(\mathcal{B}\)-**coordinates and also to** \(\mathcal{C}\)-**coordinates**.
\[
\begin{bmatrix} 1 \\ 4 \end{bmatrix}_{\mathcal{B}}, \quad
\begin{bmatrix} -2.5 \\ -0.5 \end{bmatrix}_{\mathcal{C}}
\]
Transcribed Image Text:7. Check your work by converting your answers from #1 and #2 to each other.
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