Find the vector x determined by the coordinate vector ---(DBE) 8 and the basis B = 0 XB = The vector x = The vector x = The vector x = 1 7 -3 -2 -27 H 0 5 3 2 4 []} 3
Find the vector x determined by the coordinate vector ---(DBE) 8 and the basis B = 0 XB = The vector x = The vector x = The vector x = 1 7 -3 -2 -27 H 0 5 3 2 4 []} 3
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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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}}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c4ab2a7-bf0a-4a13-a1f0-4aa79b7eca24%2F6abe93e2-0858-4eec-b7bf-0e2c1177f4e0%2Fvxorkcf_processed.jpeg&w=3840&q=75)
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}}
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