7 Find w if (w)s = |0| relative to the basis S = 3 – 10 3 W =
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
![**Problem Statement:**
Find the vector \( \mathbf{w} \) if its coordinate vector relative to the basis \( S \) is given by:
\[
[w]_S = \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}
\]
The basis \( S \) consists of the following vectors:
\[
S = \left\{ \begin{bmatrix} 7 \\ 0 \\ -10 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} -2 \\ 0 \\ 3 \end{bmatrix} \right\}
\]
Determine the vector \( \mathbf{w} \) in standard coordinates.
**Solution:**
To find \( \mathbf{w} \), we use the following expression, given that \( w = a\mathbf{v_1} + b\mathbf{v_2} + c\mathbf{v_3} \), where \( [w]_S = \begin{bmatrix} a \\ b \\ c \end{bmatrix} \):
\[
\mathbf{w} = 1 \cdot \begin{bmatrix} 7 \\ 0 \\ -10 \end{bmatrix} + 0 \cdot \begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix} + 3 \cdot \begin{bmatrix} -2 \\ 0 \\ 3 \end{bmatrix}
\]
Calculate each part:
\[
= \begin{bmatrix} 7 \\ 0 \\ -10 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} -6 \\ 0 \\ 9 \end{bmatrix}
\]
Adding these vectors gives:
\[
\mathbf{w} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}
\]
Thus, the vector \( \mathbf{w} \) in standard coordinates is:
\[
\mathbf{w} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5025798a-663c-4f82-a82a-d0677c58328a%2F7df6e24e-e0ab-41ce-8be1-bff9e1d24082%2F4sbngjs_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the vector \( \mathbf{w} \) if its coordinate vector relative to the basis \( S \) is given by:
\[
[w]_S = \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}
\]
The basis \( S \) consists of the following vectors:
\[
S = \left\{ \begin{bmatrix} 7 \\ 0 \\ -10 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} -2 \\ 0 \\ 3 \end{bmatrix} \right\}
\]
Determine the vector \( \mathbf{w} \) in standard coordinates.
**Solution:**
To find \( \mathbf{w} \), we use the following expression, given that \( w = a\mathbf{v_1} + b\mathbf{v_2} + c\mathbf{v_3} \), where \( [w]_S = \begin{bmatrix} a \\ b \\ c \end{bmatrix} \):
\[
\mathbf{w} = 1 \cdot \begin{bmatrix} 7 \\ 0 \\ -10 \end{bmatrix} + 0 \cdot \begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix} + 3 \cdot \begin{bmatrix} -2 \\ 0 \\ 3 \end{bmatrix}
\]
Calculate each part:
\[
= \begin{bmatrix} 7 \\ 0 \\ -10 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} -6 \\ 0 \\ 9 \end{bmatrix}
\]
Adding these vectors gives:
\[
\mathbf{w} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}
\]
Thus, the vector \( \mathbf{w} \) in standard coordinates is:
\[
\mathbf{w} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}
\]
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