### 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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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.

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Step 1: Define gram- Schmidt process

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