Find the vector a determined by || Bug Bounty 2 2 -408)~A -Ĥ 5 = 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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I need help with these 2 problems please help me
![The task is to find the vector \( \vec{x} \) determined by a given basis \( \mathcal{B} \) and the coordinate vector of \( \vec{x} \) with respect to \( \mathcal{B} \).
**Description of the Given Information:**
- The basis \( \mathcal{B} \) consists of three vectors:
\[
\mathcal{B} = \left\{ \begin{bmatrix} 2 \\ -6 \\ -2 \end{bmatrix}, \begin{bmatrix} -2 \\ 5 \\ 4 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} \right\}
\]
- The coordinate vector of \( \vec{x} \) with respect to the basis \( \mathcal{B} \) is:
\[
\left[ \vec{x} \right]_{\mathcal{B}} = \begin{bmatrix} -2 \\ 3 \\ 1 \end{bmatrix}
\]
- The vector \( \vec{x} \) is expressed in terms of the basis vectors and the coordinate vector.
**Solution:**
To find the vector \( \vec{x} \), perform the following calculation involving the basis vectors and the coefficients from the coordinate vector:
\[
\vec{x} = -2 \begin{bmatrix} 2 \\ -6 \\ -2 \end{bmatrix} + 3 \begin{bmatrix} -2 \\ 5 \\ 4 \end{bmatrix} + 1 \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix}
\]
**Bug Bounty:** There is a button labeled "Bug Bounty," which likely indicates an option for users to report issues or errors related to the problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa62afbd1-907f-4d39-9d6d-83ebd9acc77d%2Fb6aed0a4-3639-4357-9c2b-455ef36e8ae5%2Fajsj2qh_processed.png&w=3840&q=75)
Transcribed Image Text:The task is to find the vector \( \vec{x} \) determined by a given basis \( \mathcal{B} \) and the coordinate vector of \( \vec{x} \) with respect to \( \mathcal{B} \).
**Description of the Given Information:**
- The basis \( \mathcal{B} \) consists of three vectors:
\[
\mathcal{B} = \left\{ \begin{bmatrix} 2 \\ -6 \\ -2 \end{bmatrix}, \begin{bmatrix} -2 \\ 5 \\ 4 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} \right\}
\]
- The coordinate vector of \( \vec{x} \) with respect to the basis \( \mathcal{B} \) is:
\[
\left[ \vec{x} \right]_{\mathcal{B}} = \begin{bmatrix} -2 \\ 3 \\ 1 \end{bmatrix}
\]
- The vector \( \vec{x} \) is expressed in terms of the basis vectors and the coordinate vector.
**Solution:**
To find the vector \( \vec{x} \), perform the following calculation involving the basis vectors and the coefficients from the coordinate vector:
\[
\vec{x} = -2 \begin{bmatrix} 2 \\ -6 \\ -2 \end{bmatrix} + 3 \begin{bmatrix} -2 \\ 5 \\ 4 \end{bmatrix} + 1 \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix}
\]
**Bug Bounty:** There is a button labeled "Bug Bounty," which likely indicates an option for users to report issues or errors related to the problem.
![Find the coordinate vector, \([\vec{x}]_{\mathcal{B}}\), of \(\vec{x}\) relative to the basis \(\mathcal{B}\).
\[
\mathcal{B} = \left\{ \begin{bmatrix} 2 \\ 2 \\ 4 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 4 \\ 0 \\ 18 \end{bmatrix} \right\}, \quad \vec{x} = \begin{bmatrix} 8 \\ -6 \\ 50 \end{bmatrix}
\]
\[
[\vec{x}]_{\mathcal{B}} = \boxed{\phantom{xxx}}
\]
[Note: There is a "Bug Bounty" button below the input box.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa62afbd1-907f-4d39-9d6d-83ebd9acc77d%2Fb6aed0a4-3639-4357-9c2b-455ef36e8ae5%2F5sxkqyi_processed.png&w=3840&q=75)
Transcribed Image Text:Find the coordinate vector, \([\vec{x}]_{\mathcal{B}}\), of \(\vec{x}\) relative to the basis \(\mathcal{B}\).
\[
\mathcal{B} = \left\{ \begin{bmatrix} 2 \\ 2 \\ 4 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 4 \\ 0 \\ 18 \end{bmatrix} \right\}, \quad \vec{x} = \begin{bmatrix} 8 \\ -6 \\ 50 \end{bmatrix}
\]
\[
[\vec{x}]_{\mathcal{B}} = \boxed{\phantom{xxx}}
\]
[Note: There is a "Bug Bounty" button below the input box.]
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