What vector v is given by the coordinate vector 3 [1] 6 B={[-3 8 7], [7 1 6], [7 -4 9]}. ?
What vector v is given by the coordinate vector 3 [1] 6 B={[-3 8 7], [7 1 6], [7 -4 9]}. ?
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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Question
![**Vector Representation Using a Basis**
This section explores how to determine a vector \( \mathbf{v} \) given its coordinate vector with respect to a specific basis.
### Problem Statement
We are given a coordinate vector:
\[
\begin{bmatrix}
6 \\
3 \\
6
\end{bmatrix}_{\mathcal{B}}
\]
Our task is to find the vector \( \mathbf{v} \) associated with this coordinate vector based on the specified basis \( \mathcal{B} \).
### Basis \( \mathcal{B} \)
The basis \(\mathcal{B}\) is defined as:
\[
\mathcal{B} = \left\{
\begin{bmatrix} -3 \\ 8 \\ 7 \end{bmatrix},
\begin{bmatrix} 7 \\ 1 \\ 6 \end{bmatrix},
\begin{bmatrix} 7 \\ -4 \\ 9 \end{bmatrix}
\right\}
\]
### Explanation
To find the vector \( \mathbf{v} \), we use the coordinate vector and the given basis, applying the concept that the coordinate vector represents a linear combination of the basis vectors.
\[
\mathbf{v} = 6 \begin{bmatrix} -3 \\ 8 \\ 7 \end{bmatrix} + 3 \begin{bmatrix} 7 \\ 1 \\ 6 \end{bmatrix} + 6 \begin{bmatrix} 7 \\ -4 \\ 9 \end{bmatrix}
\]
By calculating this expression, we determine the vector \( \mathbf{v} \) in the standard coordinate system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86f92849-52f1-43f0-ad77-c1523309cdc1%2F8524f327-d5ac-43e0-93bb-210702191d4c%2Fyhauy97_processed.png&w=3840&q=75)
Transcribed Image Text:**Vector Representation Using a Basis**
This section explores how to determine a vector \( \mathbf{v} \) given its coordinate vector with respect to a specific basis.
### Problem Statement
We are given a coordinate vector:
\[
\begin{bmatrix}
6 \\
3 \\
6
\end{bmatrix}_{\mathcal{B}}
\]
Our task is to find the vector \( \mathbf{v} \) associated with this coordinate vector based on the specified basis \( \mathcal{B} \).
### Basis \( \mathcal{B} \)
The basis \(\mathcal{B}\) is defined as:
\[
\mathcal{B} = \left\{
\begin{bmatrix} -3 \\ 8 \\ 7 \end{bmatrix},
\begin{bmatrix} 7 \\ 1 \\ 6 \end{bmatrix},
\begin{bmatrix} 7 \\ -4 \\ 9 \end{bmatrix}
\right\}
\]
### Explanation
To find the vector \( \mathbf{v} \), we use the coordinate vector and the given basis, applying the concept that the coordinate vector represents a linear combination of the basis vectors.
\[
\mathbf{v} = 6 \begin{bmatrix} -3 \\ 8 \\ 7 \end{bmatrix} + 3 \begin{bmatrix} 7 \\ 1 \\ 6 \end{bmatrix} + 6 \begin{bmatrix} 7 \\ -4 \\ 9 \end{bmatrix}
\]
By calculating this expression, we determine the vector \( \mathbf{v} \) in the standard coordinate system.
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