6 What vector v is given by the coordinate vector 3 ? 6 B = { [−3 8 7], [7 1 6], [7 −4 9]}. B
6 What vector v is given by the coordinate vector 3 ? 6 B = { [−3 8 7], [7 1 6], [7 −4 9]}. 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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Question
![The problem posed is: "What vector **v** is given by the coordinate vector \(\begin{bmatrix} 6 \\ 3 \\ 6 \end{bmatrix}_{\mathcal{B}} \)?"
The basis \(\mathcal{B}\) is defined as follows:
\[
\mathcal{B} = \left\{ \begin{bmatrix} -3 \\ 8 \\ 7 \end{bmatrix}, \begin{bmatrix} 7 \\ 1 \\ 6 \end{bmatrix}, \begin{bmatrix} 7 \\ -4 \\ 9 \end{bmatrix} \right\}.
\]
To find the vector **v** in standard coordinates, you must express **v** as a linear combination of the basis vectors given by \(\mathcal{B}\) using the coefficients in the coordinate vector. This means calculating:
\[
\textbf{v} = 6 \begin{bmatrix} -3 \\ 8 \\ 7 \end{bmatrix} + 3 \begin{bmatrix} 7 \\ 1 \\ 6 \end{bmatrix} + 6 \begin{bmatrix} 7 \\ -4 \\ 9 \end{bmatrix}
\]
Calculate each product and sum them to find the vector **v**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b87c16f-5797-4fee-9867-0be2bea00aaa%2Fb97840a8-a472-4dd8-b7e6-f6ee1556941b%2Fteu1zol_processed.png&w=3840&q=75)
Transcribed Image Text:The problem posed is: "What vector **v** is given by the coordinate vector \(\begin{bmatrix} 6 \\ 3 \\ 6 \end{bmatrix}_{\mathcal{B}} \)?"
The basis \(\mathcal{B}\) is defined as follows:
\[
\mathcal{B} = \left\{ \begin{bmatrix} -3 \\ 8 \\ 7 \end{bmatrix}, \begin{bmatrix} 7 \\ 1 \\ 6 \end{bmatrix}, \begin{bmatrix} 7 \\ -4 \\ 9 \end{bmatrix} \right\}.
\]
To find the vector **v** in standard coordinates, you must express **v** as a linear combination of the basis vectors given by \(\mathcal{B}\) using the coefficients in the coordinate vector. This means calculating:
\[
\textbf{v} = 6 \begin{bmatrix} -3 \\ 8 \\ 7 \end{bmatrix} + 3 \begin{bmatrix} 7 \\ 1 \\ 6 \end{bmatrix} + 6 \begin{bmatrix} 7 \\ -4 \\ 9 \end{bmatrix}
\]
Calculate each product and sum them to find the vector **v**.
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