Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
expand_more
expand_more
format_list_bulleted
Related questions
Question
Solve this mutliple choice
![**Finding the Vector \( \mathbf{x} \) Based on Coordinate Vector \( \mathbf{x}_{\mathcal{B}} \)**
The task is to determine the vector \( \mathbf{x} \) based on the given coordinate vector \( \mathbf{x}_{\mathcal{B}} \) and the basis \( \mathcal{B} \).
Given data:
\[ \mathbf{x}_{\mathcal{B}} = \begin{bmatrix} -4 \\ 8 \\ -7 \end{bmatrix} \]
and the basis \( \mathcal{B} \) is:
\[ \mathcal{B} = \left\{ \begin{bmatrix} -1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ -5 \\ 2 \end{bmatrix}, \begin{bmatrix} 4 \\ -7 \\ 3 \end{bmatrix} \right\} \]
Next, we need to identify the vector \( \mathbf{x} \) from the given multiple choice options.
### Options Provided:
1. \(\mathbf{x} = \begin{bmatrix} 0 \\ 1 \\ -5 \end{bmatrix}\)
2. \(\mathbf{x} = \begin{bmatrix} 7 \\ -3 \\ -2 \end{bmatrix}\) <span style="background-color: #E6E6FA;">(Selected)</span>
3. \(\mathbf{x} = \begin{bmatrix} -2 \\ 0 \\ 5 \end{bmatrix}\)
### Explanation of the Provided Basis and Coordinate Vector
A basis in linear algebra is a set of linearly independent vectors in a vector space such that any vector in the space can be expressed as a linear combination of these basis vectors. The given basis consists of three vectors:
\[ \begin{bmatrix} -1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ -5 \\ 2 \end{bmatrix}, \begin{bmatrix} 4 \\ -7 \\ 3 \end{bmatrix} \]
Given that these vectors form a basis for the space, the vector \( \mathbf{x} \) can be written as a linear combination of these basis vectors, weighted by the coefficients in \( \mathbf{x}_{\mathcal{B}}](https://content.bartleby.com/qna-images/question/0c4ab2a7-bf0a-4a13-a1f0-4aa79b7eca24/6abe93e2-0858-4eec-b7bf-0e2c1177f4e0/vxorkcf_processed.jpeg)
Transcribed Image Text:**Finding the Vector \( \mathbf{x} \) Based on Coordinate Vector \( \mathbf{x}_{\mathcal{B}} \)**
The task is to determine the vector \( \mathbf{x} \) based on the given coordinate vector \( \mathbf{x}_{\mathcal{B}} \) and the basis \( \mathcal{B} \).
Given data:
\[ \mathbf{x}_{\mathcal{B}} = \begin{bmatrix} -4 \\ 8 \\ -7 \end{bmatrix} \]
and the basis \( \mathcal{B} \) is:
\[ \mathcal{B} = \left\{ \begin{bmatrix} -1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ -5 \\ 2 \end{bmatrix}, \begin{bmatrix} 4 \\ -7 \\ 3 \end{bmatrix} \right\} \]
Next, we need to identify the vector \( \mathbf{x} \) from the given multiple choice options.
### Options Provided:
1. \(\mathbf{x} = \begin{bmatrix} 0 \\ 1 \\ -5 \end{bmatrix}\)
2. \(\mathbf{x} = \begin{bmatrix} 7 \\ -3 \\ -2 \end{bmatrix}\) <span style="background-color: #E6E6FA;">(Selected)</span>
3. \(\mathbf{x} = \begin{bmatrix} -2 \\ 0 \\ 5 \end{bmatrix}\)
### Explanation of the Provided Basis and Coordinate Vector
A basis in linear algebra is a set of linearly independent vectors in a vector space such that any vector in the space can be expressed as a linear combination of these basis vectors. The given basis consists of three vectors:
\[ \begin{bmatrix} -1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ -5 \\ 2 \end{bmatrix}, \begin{bmatrix} 4 \\ -7 \\ 3 \end{bmatrix} \]
Given that these vectors form a basis for the space, the vector \( \mathbf{x} \) can be written as a linear combination of these basis vectors, weighted by the coefficients in \( \mathbf{x}_{\mathcal{B}}
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by stepSolved in 3 steps with 3 images
Knowledge Booster
Similar questions
Recommended textbooks for you
- Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
Numerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill Education
Introductory Mathematics for Engineering Applicat...Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEY 
Mathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,
Advanced Engineering Mathematics
Advanced Math
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:9780073397924
Author:Steven C. Chapra Dr., Raymond P. Canale
Publisher:McGraw-Hill Education
Introductory Mathematics for Engineering Applicat...
Advanced Math
ISBN:9781118141809
Author:Nathan Klingbeil
Publisher:WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,