4. y = 3, U₁ 3 4, 12 3 0
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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![### Linear Algebra Exercises on Orthogonal Sets
In the following exercises (3-6), we are asked to verify that the set \(\{ \mathbf{u_1}, \mathbf{u_2} \}\) is an orthogonal set. After verification, we will find the orthogonal projection of the vector \(\mathbf{y}\) onto the span of \(\{ \mathbf{u_1}, \mathbf{u_2} \}\).
#### Exercise 3:
Given vectors:
\[ \mathbf{y} = \begin{bmatrix} -1 \\ 4 \\ 3 \end{bmatrix}, \quad \mathbf{u_1} = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \quad \mathbf{u_2} = \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix} \]
#### Exercise 4:
Given vectors:
\[ \mathbf{y} = \begin{bmatrix} 4 \\ 3 \\ -2 \end{bmatrix}, \quad \mathbf{u_1} = \begin{bmatrix} 3 \\ 4 \\ 0 \end{bmatrix}, \quad \mathbf{u_2} = \begin{bmatrix} -4 \\ 3 \\ 0 \end{bmatrix} \]
### Instructions:
1. **Verification of Orthogonality:**
- To verify that \(\{ \mathbf{u_1}, \mathbf{u_2} \}\) is an orthogonal set, compute the dot product \(\mathbf{u_1} \cdot \mathbf{u_2}\). If the dot product is zero, the vectors are orthogonal.
2. **Finding Orthogonal Projections:**
- To find the orthogonal projection of \(\mathbf{y}\) onto \(\text{Span}\{ \mathbf{u_1}, \mathbf{u_2} \}\), use the formula for orthogonal projection:
\[
\text{proj}_{\mathbf{u}} \mathbf{y} = \frac{\mathbf{u} \cdot \mathbf{y}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}
\]
- Apply this formula for both \(\mathbf{u_1}\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F324f16c4-c4e5-4b06-b074-f261b39e021a%2F1fd2fc47-55ac-48ee-81c0-ec7c249b993f%2F5tbjdjg_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Exercises on Orthogonal Sets
In the following exercises (3-6), we are asked to verify that the set \(\{ \mathbf{u_1}, \mathbf{u_2} \}\) is an orthogonal set. After verification, we will find the orthogonal projection of the vector \(\mathbf{y}\) onto the span of \(\{ \mathbf{u_1}, \mathbf{u_2} \}\).
#### Exercise 3:
Given vectors:
\[ \mathbf{y} = \begin{bmatrix} -1 \\ 4 \\ 3 \end{bmatrix}, \quad \mathbf{u_1} = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \quad \mathbf{u_2} = \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix} \]
#### Exercise 4:
Given vectors:
\[ \mathbf{y} = \begin{bmatrix} 4 \\ 3 \\ -2 \end{bmatrix}, \quad \mathbf{u_1} = \begin{bmatrix} 3 \\ 4 \\ 0 \end{bmatrix}, \quad \mathbf{u_2} = \begin{bmatrix} -4 \\ 3 \\ 0 \end{bmatrix} \]
### Instructions:
1. **Verification of Orthogonality:**
- To verify that \(\{ \mathbf{u_1}, \mathbf{u_2} \}\) is an orthogonal set, compute the dot product \(\mathbf{u_1} \cdot \mathbf{u_2}\). If the dot product is zero, the vectors are orthogonal.
2. **Finding Orthogonal Projections:**
- To find the orthogonal projection of \(\mathbf{y}\) onto \(\text{Span}\{ \mathbf{u_1}, \mathbf{u_2} \}\), use the formula for orthogonal projection:
\[
\text{proj}_{\mathbf{u}} \mathbf{y} = \frac{\mathbf{u} \cdot \mathbf{y}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}
\]
- Apply this formula for both \(\mathbf{u_1}\
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