What vector v is given by the coordinate vector B = { 9x +4, 4x + 6 }. [3] ? 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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![### Problem Statement
**Question**: What vector **v** is given by the coordinate vector
\[
\begin{bmatrix} 8 \\ -5 \end{bmatrix}_{\mathcal{B}}
\]
where
\[
\mathcal{B} = \{ 9x + 4, \; 4x + 6 \}.
\]
### Explanation
In this problem, we are working with a coordinate vector expressed with respect to a basis \(\mathcal{B}\). The vector \(\begin{bmatrix} 8 \\ -5 \end{bmatrix}_{\mathcal{B}}\) indicates that the vector **v** can be expressed as a linear combination of the basis vectors in \(\mathcal{B}\). The basis \(\mathcal{B}\) consists of the functions \(9x + 4\) and \(4x + 6\).
To find the vector **v**, you should calculate the linear combination:
\[
v = 8 \cdot (9x + 4) + (-5) \cdot (4x + 6)
\]
### Steps to Solve
1. **Multiply each basis vector by its corresponding scalar**:
- \(8 \cdot (9x + 4) = 72x + 32\)
- \(-5 \cdot (4x + 6) = -20x - 30\)
2. **Add the resulting expressions**:
- \(v = (72x + 32) + (-20x - 30)\)
3. **Combine like terms**:
- \(v = (72x - 20x) + (32 - 30)\)
- \(v = 52x + 2\)
Thus, the vector **v** is \(52x + 2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b87c16f-5797-4fee-9867-0be2bea00aaa%2F5079adfc-1645-4281-bf23-2fd66fbac608%2F97lgue_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Question**: What vector **v** is given by the coordinate vector
\[
\begin{bmatrix} 8 \\ -5 \end{bmatrix}_{\mathcal{B}}
\]
where
\[
\mathcal{B} = \{ 9x + 4, \; 4x + 6 \}.
\]
### Explanation
In this problem, we are working with a coordinate vector expressed with respect to a basis \(\mathcal{B}\). The vector \(\begin{bmatrix} 8 \\ -5 \end{bmatrix}_{\mathcal{B}}\) indicates that the vector **v** can be expressed as a linear combination of the basis vectors in \(\mathcal{B}\). The basis \(\mathcal{B}\) consists of the functions \(9x + 4\) and \(4x + 6\).
To find the vector **v**, you should calculate the linear combination:
\[
v = 8 \cdot (9x + 4) + (-5) \cdot (4x + 6)
\]
### Steps to Solve
1. **Multiply each basis vector by its corresponding scalar**:
- \(8 \cdot (9x + 4) = 72x + 32\)
- \(-5 \cdot (4x + 6) = -20x - 30\)
2. **Add the resulting expressions**:
- \(v = (72x + 32) + (-20x - 30)\)
3. **Combine like terms**:
- \(v = (72x - 20x) + (32 - 30)\)
- \(v = 52x + 2\)
Thus, the vector **v** is \(52x + 2\).
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