Use the Gram-Schmidt process to transform the basis ü, = (1,0,0),ü, = (3,7,-2),ū; = (0,4,1) into orthogonal basis. %3D
Use the Gram-Schmidt process to transform the basis ü, = (1,0,0),ü, = (3,7,-2),ū; = (0,4,1) into orthogonal basis. %3D
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: Using the Gram-Schmidt Process for Orthogonalization**
**Objective:**
Learn how to use the Gram-Schmidt process to transform a set of vectors into an orthogonal basis.
**Given Vectors:**
- \(\vec{u_1} = (1, 0, 0)\)
- \(\vec{u_2} = (3, 7, -2)\)
- \(\vec{u_3} = (0, 4, 1)\)
**Task:**
Transform the given basis into an orthogonal basis using the Gram-Schmidt process.
**Instructions:**
1. Begin with the first vector \(\vec{u_1}\). It remains unchanged as \(\vec{v_1} = \vec{u_1}\).
2. Apply the Gram-Schmidt process to find \(\vec{v_2}\) by orthogonalizing \(\vec{u_2}\) against \(\vec{v_1}\):
\[
\vec{v_2} = \vec{u_2} - \text{proj}_{\vec{v_1}}(\vec{u_2})
\]
3. Continue the process for \(\vec{u_3}\) to find \(\vec{v_3}\):
\[
\vec{v_3} = \vec{u_3} - \text{proj}_{\vec{v_1}}(\vec{u_3}) - \text{proj}_{\vec{v_2}}(\vec{u_3})
\]
5. Verify that \(\vec{v_1}\), \(\vec{v_2}\), and \(\vec{v_3}\) are orthogonal by ensuring the dot products are zero.
This lesson illustrates the step-by-step Gram-Schmidt orthogonalization process to convert a basis into an orthogonal set, providing a foundation for further mathematical applications in vector spaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2520f39-8b3c-4b2b-be15-831ac0e5241f%2F0aa22ca2-4cc6-4e39-8101-a1b42ac82d76%2Fxivyn4y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Using the Gram-Schmidt Process for Orthogonalization**
**Objective:**
Learn how to use the Gram-Schmidt process to transform a set of vectors into an orthogonal basis.
**Given Vectors:**
- \(\vec{u_1} = (1, 0, 0)\)
- \(\vec{u_2} = (3, 7, -2)\)
- \(\vec{u_3} = (0, 4, 1)\)
**Task:**
Transform the given basis into an orthogonal basis using the Gram-Schmidt process.
**Instructions:**
1. Begin with the first vector \(\vec{u_1}\). It remains unchanged as \(\vec{v_1} = \vec{u_1}\).
2. Apply the Gram-Schmidt process to find \(\vec{v_2}\) by orthogonalizing \(\vec{u_2}\) against \(\vec{v_1}\):
\[
\vec{v_2} = \vec{u_2} - \text{proj}_{\vec{v_1}}(\vec{u_2})
\]
3. Continue the process for \(\vec{u_3}\) to find \(\vec{v_3}\):
\[
\vec{v_3} = \vec{u_3} - \text{proj}_{\vec{v_1}}(\vec{u_3}) - \text{proj}_{\vec{v_2}}(\vec{u_3})
\]
5. Verify that \(\vec{v_1}\), \(\vec{v_2}\), and \(\vec{v_3}\) are orthogonal by ensuring the dot products are zero.
This lesson illustrates the step-by-step Gram-Schmidt orthogonalization process to convert a basis into an orthogonal set, providing a foundation for further mathematical applications in vector spaces.
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