Orthonormalize using Gram-Schmidt. -8 15 {[][]} ([181181)
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: Orthonormalization Using the Gram-Schmidt Process**
**Objective:**
Orthonormalize the given set of vectors using the Gram-Schmidt process.
**Vectors to Orthonormalize:**
The initial set of vectors provided is:
\[
\left\{ \begin{bmatrix} -8 \\ 4 \end{bmatrix}, \begin{bmatrix} 15 \\ 4 \end{bmatrix} \right\}
\]
**Process:**
1. **Gram-Schmidt Algorithm Overview:**
- The Gram-Schmidt process is a method for orthonormalizing a set of vectors in an inner product space.
- The process converts a set of linearly independent vectors into a set of orthogonal vectors, then normalizes them.
2. **Calculation Steps:**
- Begin with the first vector, normalize it.
- Orthogonalize the second vector with respect to the first, and normalize.
**Resulting Orthonormal Set:**
After completing the Gram-Schmidt process, the orthonormal set will be:
\[
\left\{ \begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix}, \begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix} \right\}
\]
*Note: The boxed areas indicate where the calculated orthonormal components of the vectors should appear after the computation.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F807728d5-81cc-4dac-b6a1-9785980c085b%2F5fa34687-fd52-4e61-ae43-d8a7667286ca%2F4ntizbj_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Orthonormalization Using the Gram-Schmidt Process**
**Objective:**
Orthonormalize the given set of vectors using the Gram-Schmidt process.
**Vectors to Orthonormalize:**
The initial set of vectors provided is:
\[
\left\{ \begin{bmatrix} -8 \\ 4 \end{bmatrix}, \begin{bmatrix} 15 \\ 4 \end{bmatrix} \right\}
\]
**Process:**
1. **Gram-Schmidt Algorithm Overview:**
- The Gram-Schmidt process is a method for orthonormalizing a set of vectors in an inner product space.
- The process converts a set of linearly independent vectors into a set of orthogonal vectors, then normalizes them.
2. **Calculation Steps:**
- Begin with the first vector, normalize it.
- Orthogonalize the second vector with respect to the first, and normalize.
**Resulting Orthonormal Set:**
After completing the Gram-Schmidt process, the orthonormal set will be:
\[
\left\{ \begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix}, \begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix} \right\}
\]
*Note: The boxed areas indicate where the calculated orthonormal components of the vectors should appear after the computation.*
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