In Exercises 7-10, let W be the subspace spanned by the u's, and write y as the sum of a vector in W and a vector orthogonal to W. 7. y = 3, u₁ = 5 3, U₂ = 5 1
In Exercises 7-10, let W be the subspace spanned by the u's, and write y as the sum of a vector in W and a vector orthogonal to W. 7. y = 3, u₁ = 5 3, U₂ = 5 1
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 Subspaces and Orthogonality
In **Exercises 7–10**, let \( W \) be the subspace spanned by the \( \mathbf{u} \) vectors, and express \( \mathbf{y} \) as the sum of a vector in \( W \) and a vector orthogonal to \( W \).
**Exercise 7:**
Given vectors:
\[ \mathbf{y} = \begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}, \quad \mathbf{u_1} = \begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}, \quad \mathbf{u_2} = \begin{bmatrix} 1 \\ 3 \\ -2 \end{bmatrix} \]
Find the sum representation:
\[ \mathbf{y} = \mathbf{v} + \mathbf{w} \]
where \( \mathbf{v} \in W \) and \( \mathbf{w} \perp W \).
**Exercise 8:**
Given vectors:
\[ \mathbf{y} = \begin{bmatrix} -1 \\ 4 \\ 3 \end{bmatrix}, \quad \mathbf{u_1} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{u_2} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \]
Find the sum representation:
\[ \mathbf{y} = \begin{bmatrix} -1 \\ 3 \\ -2 \end{bmatrix} \]
---
In these exercises, students are asked to decompose the vector \( \mathbf{y} \) into two components: one that lies within the subspace \( W \), which is spanned by the given vectors \( \mathbf{u_1} \) and \( \mathbf{u_2} \), and one that is orthogonal to this subspace.
### Explanation
1. **Identify Subspace \( W \)**:
- \( W \) is the subspace spanned by the vectors \( \mathbf{u_1} \) and \( \mathbf{u_2} \).
2. **Orthogonal Components**:
- The goal is to write \( \mathbf{y](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F324f16c4-c4e5-4b06-b074-f261b39e021a%2F21e39811-fca6-4401-a5c8-c7cb81831db6%2Fqjkj9lr_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Exercises on Subspaces and Orthogonality
In **Exercises 7–10**, let \( W \) be the subspace spanned by the \( \mathbf{u} \) vectors, and express \( \mathbf{y} \) as the sum of a vector in \( W \) and a vector orthogonal to \( W \).
**Exercise 7:**
Given vectors:
\[ \mathbf{y} = \begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}, \quad \mathbf{u_1} = \begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}, \quad \mathbf{u_2} = \begin{bmatrix} 1 \\ 3 \\ -2 \end{bmatrix} \]
Find the sum representation:
\[ \mathbf{y} = \mathbf{v} + \mathbf{w} \]
where \( \mathbf{v} \in W \) and \( \mathbf{w} \perp W \).
**Exercise 8:**
Given vectors:
\[ \mathbf{y} = \begin{bmatrix} -1 \\ 4 \\ 3 \end{bmatrix}, \quad \mathbf{u_1} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{u_2} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \]
Find the sum representation:
\[ \mathbf{y} = \begin{bmatrix} -1 \\ 3 \\ -2 \end{bmatrix} \]
---
In these exercises, students are asked to decompose the vector \( \mathbf{y} \) into two components: one that lies within the subspace \( W \), which is spanned by the given vectors \( \mathbf{u_1} \) and \( \mathbf{u_2} \), and one that is orthogonal to this subspace.
### Explanation
1. **Identify Subspace \( W \)**:
- \( W \) is the subspace spanned by the vectors \( \mathbf{u_1} \) and \( \mathbf{u_2} \).
2. **Orthogonal Components**:
- The goal is to write \( \mathbf{y
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