The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W. An orthogonal basis for W is. (Type a vector or list of vectors. Use a comma to separate vectors as needed.) 9 04 2 - 2 N
The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W. An orthogonal basis for W is. (Type a vector or list of vectors. Use a comma to separate vectors as needed.) 9 04 2 - 2 N
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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Question
![### Orthogonal Basis using the Gram-Schmidt Process
#### Problem Statement:
The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.
The basis set provided is:
\[
\left \{
\begin{bmatrix}
0 \\
2 \\
2
\end{bmatrix},
\begin{bmatrix}
9 \\
4 \\
-2
\end{bmatrix}
\right \}
\]
---
#### Solution:
An orthogonal basis for \( W \) is:
\[ \left \{ \boxed{\phantom{...}} \right \} \]
(Type a vector or list of vectors. Use a comma to separate vectors as needed.)
---
#### Notes:
- The Gram-Schmidt process is a method for orthogonalizing a set of vectors in an inner product space, which most commonly means Euclidean space.
- Begin with the original basis vectors and apply the Gram-Schmidt formula to find the orthogonal vectors.
- Ensure the orthogonality by verifying that the inner product (dot product) of each pair of resulting vectors is zero.
Note: The image provides a clear layout of the problem and the basis vectors to be used in the Gram-Schmidt process.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5246d2a-5796-4784-a543-e596d3b5542c%2F33ca751a-85fc-468c-9e85-27dc5d4a36c6%2Fwmy9008_processed.png&w=3840&q=75)
Transcribed Image Text:### Orthogonal Basis using the Gram-Schmidt Process
#### Problem Statement:
The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.
The basis set provided is:
\[
\left \{
\begin{bmatrix}
0 \\
2 \\
2
\end{bmatrix},
\begin{bmatrix}
9 \\
4 \\
-2
\end{bmatrix}
\right \}
\]
---
#### Solution:
An orthogonal basis for \( W \) is:
\[ \left \{ \boxed{\phantom{...}} \right \} \]
(Type a vector or list of vectors. Use a comma to separate vectors as needed.)
---
#### Notes:
- The Gram-Schmidt process is a method for orthogonalizing a set of vectors in an inner product space, which most commonly means Euclidean space.
- Begin with the original basis vectors and apply the Gram-Schmidt formula to find the orthogonal vectors.
- Ensure the orthogonality by verifying that the inner product (dot product) of each pair of resulting vectors is zero.
Note: The image provides a clear layout of the problem and the basis vectors to be used in the Gram-Schmidt process.
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