The set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W. Assume the vectors are in the order x, and x,. 1 3 1 2 - 7 1 1 The orthogonal basis produced using the Gram-Schmidt process for W is { }. (Use a comma to separate vectors as needed.)
The set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W. Assume the vectors are in the order x, and x,. 1 3 1 2 - 7 1 1 The orthogonal basis produced using the Gram-Schmidt process for W is { }. (Use a comma to separate vectors as needed.)
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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![### Orthogonal Basis Using Gram-Schmidt Process
**Objective:**
To find the orthogonal basis for a subspace \( W \) using the Gram-Schmidt process.
**Given:**
The following set of vectors form a basis for a subspace \( W \):
\[
\mathbf{x_1} =
\begin{pmatrix}
1 \\
-1 \\
0 \\
1
\end{pmatrix}
, \quad
\mathbf{x_2} =
\begin{pmatrix}
3 \\
-2 \\
-7 \\
1
\end{pmatrix}
\]
**Step-by-Step Process:**
1. **Starting with \(\mathbf{x_1}\):**
Set \(\mathbf{u_1} = \mathbf{x_1}\).
2. **Making \(\mathbf{x_2}\) orthogonal to \(\mathbf{u_1}\):**
Compute the projection of \(\mathbf{x_2}\) onto \(\mathbf{u_1}\):
\[
\text{proj}_{\mathbf{u_1}}\mathbf{x_2} = \frac{\mathbf{x_2} \cdot \mathbf{u_1}}{\mathbf{u_1} \cdot \mathbf{u_1}} \mathbf{u_1}
\]
Subtract this projection from \(\mathbf{x_2}\) to get \(\mathbf{u_2}\):
\[
\mathbf{u_2} = \mathbf{x_2} - \text{proj}_{\mathbf{u_1}}\mathbf{x_2}
\]
**Outcome:**
The orthogonal basis produced using the Gram-Schmidt process for \( W \) is as follows:
\[
\mathbf{u_1} =
\begin{pmatrix}
1 \\
-1 \\
0 \\
1
\end{pmatrix}
, \quad
\mathbf{u_2} = \text{(Calculated as per the steps above)}
\]
For the actual values of \(\mathbf{u_2}\), you would proceed with the calculation steps explicitly to find the specific orthogonal vectors.
**Note:** Remember to use a comma to separate the vectors when listing the orthogonal basis vectors.
---
This example illustrates the application of the Gram-Schmidt process to transform a basis into an orthogonal basis, enhancing understanding for students studying linear algebra concepts on an](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05b5783f-5507-4009-950e-cf9b7485abe1%2F8be42c98-5776-44ce-96d1-cb07b2e24a61%2F314xbjl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Orthogonal Basis Using Gram-Schmidt Process
**Objective:**
To find the orthogonal basis for a subspace \( W \) using the Gram-Schmidt process.
**Given:**
The following set of vectors form a basis for a subspace \( W \):
\[
\mathbf{x_1} =
\begin{pmatrix}
1 \\
-1 \\
0 \\
1
\end{pmatrix}
, \quad
\mathbf{x_2} =
\begin{pmatrix}
3 \\
-2 \\
-7 \\
1
\end{pmatrix}
\]
**Step-by-Step Process:**
1. **Starting with \(\mathbf{x_1}\):**
Set \(\mathbf{u_1} = \mathbf{x_1}\).
2. **Making \(\mathbf{x_2}\) orthogonal to \(\mathbf{u_1}\):**
Compute the projection of \(\mathbf{x_2}\) onto \(\mathbf{u_1}\):
\[
\text{proj}_{\mathbf{u_1}}\mathbf{x_2} = \frac{\mathbf{x_2} \cdot \mathbf{u_1}}{\mathbf{u_1} \cdot \mathbf{u_1}} \mathbf{u_1}
\]
Subtract this projection from \(\mathbf{x_2}\) to get \(\mathbf{u_2}\):
\[
\mathbf{u_2} = \mathbf{x_2} - \text{proj}_{\mathbf{u_1}}\mathbf{x_2}
\]
**Outcome:**
The orthogonal basis produced using the Gram-Schmidt process for \( W \) is as follows:
\[
\mathbf{u_1} =
\begin{pmatrix}
1 \\
-1 \\
0 \\
1
\end{pmatrix}
, \quad
\mathbf{u_2} = \text{(Calculated as per the steps above)}
\]
For the actual values of \(\mathbf{u_2}\), you would proceed with the calculation steps explicitly to find the specific orthogonal vectors.
**Note:** Remember to use a comma to separate the vectors when listing the orthogonal basis vectors.
---
This example illustrates the application of the Gram-Schmidt process to transform a basis into an orthogonal basis, enhancing understanding for students studying linear algebra concepts on an
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