Find an orthogonal set with the same span (and same number of elements) as 9 27 12 11 9 12 11 -9 13 -10 21 -36 X 1026 223 2200 223 84 223 1809 223 0000
Find an orthogonal set with the same span (and same number of elements) as 9 27 12 11 9 12 11 -9 13 -10 21 -36 X 1026 223 2200 223 84 223 1809 223 0000
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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![**Title: Finding an Orthogonal Set with the Same Span**
**Goal:**
Find an orthogonal set with the same span (and same number of elements) as the given vectors.
**Given Vectors:**
\[
\left\{
\begin{bmatrix}
9 \\
12 \\
11 \\
-10
\end{bmatrix},
\begin{bmatrix}
-9 \\
4 \\
-5 \\
13
\end{bmatrix},
\begin{bmatrix}
27 \\
4 \\
21 \\
-36
\end{bmatrix}
\right\}
\]
**Step-by-step Process:**
1. **First Vector:**
\[
\begin{bmatrix}
9 \\
12 \\
11 \\
-10
\end{bmatrix}
\]
2. **Second Vector:**
Orthogonal component calculation:
\[
\begin{bmatrix}
-\frac{1026}{223} \\
\frac{2200}{223} \\
\frac{84}{223} \\
\frac{1809}{223}
\end{bmatrix}
\]
3. **Third Vector:**
Orthogonal component yet to be determined:
\[
\begin{bmatrix}
? \\
? \\
? \\
?
\end{bmatrix}
\]
**Explanation of Graphical Elements:**
- **Braces and Brackets:**
Define the set of vectors we are working with. Each vector is enclosed in a bracket, and they are collectively grouped with a brace.
- **Boxes:**
Each component of the vectors is presented within individual boxes to clearly distinguish them.
- **Calculation of Orthogonal Components:**
The middle vector's components are expressed in terms of fractions, indicating the result of orthogonalization (potentially using methods like the Gram-Schmidt process).
**Conclusion:**
The objective is to identify a set of vectors that are orthogonal to each other while maintaining the same span as the original set. The calculations after the first vector aim to orthogonalize subsequent vectors with respect to those already processed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F807728d5-81cc-4dac-b6a1-9785980c085b%2Fe0178a3c-bcb9-43b5-ae4e-b93d6db47628%2F5gyag9d_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Finding an Orthogonal Set with the Same Span**
**Goal:**
Find an orthogonal set with the same span (and same number of elements) as the given vectors.
**Given Vectors:**
\[
\left\{
\begin{bmatrix}
9 \\
12 \\
11 \\
-10
\end{bmatrix},
\begin{bmatrix}
-9 \\
4 \\
-5 \\
13
\end{bmatrix},
\begin{bmatrix}
27 \\
4 \\
21 \\
-36
\end{bmatrix}
\right\}
\]
**Step-by-step Process:**
1. **First Vector:**
\[
\begin{bmatrix}
9 \\
12 \\
11 \\
-10
\end{bmatrix}
\]
2. **Second Vector:**
Orthogonal component calculation:
\[
\begin{bmatrix}
-\frac{1026}{223} \\
\frac{2200}{223} \\
\frac{84}{223} \\
\frac{1809}{223}
\end{bmatrix}
\]
3. **Third Vector:**
Orthogonal component yet to be determined:
\[
\begin{bmatrix}
? \\
? \\
? \\
?
\end{bmatrix}
\]
**Explanation of Graphical Elements:**
- **Braces and Brackets:**
Define the set of vectors we are working with. Each vector is enclosed in a bracket, and they are collectively grouped with a brace.
- **Boxes:**
Each component of the vectors is presented within individual boxes to clearly distinguish them.
- **Calculation of Orthogonal Components:**
The middle vector's components are expressed in terms of fractions, indicating the result of orthogonalization (potentially using methods like the Gram-Schmidt process).
**Conclusion:**
The objective is to identify a set of vectors that are orthogonal to each other while maintaining the same span as the original set. The calculations after the first vector aim to orthogonalize subsequent vectors with respect to those already processed.
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