Find the orthogonal projection y of the vector y = 6 -2 4 2 onto the subspace W = Span u= 2 2
Find the orthogonal projection y of the vector y = 6 -2 4 2 onto the subspace W = Span u= 2 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Orthogonal Complements
### Section 7.5.2: Orthogonal Projections
#### Challenge Activity
**Problem Statement:**
Find the orthogonal projection \(\hat{y}\) of the vector \(\mathbf{y} = \begin{bmatrix} 6 \\ -2 \\ 2 \end{bmatrix}\) onto the subspace \(W = \text{Span} \left\{ \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix} \right\} \).
\[
\hat{y} =
\begin{bmatrix}
\text{Enter value 1} \\
\text{Enter value 2} \\
\text{Enter value 3}
\end{bmatrix}
\]
**Input Fields:**
- Enter value 1: Textbox
- Enter value 2: Textbox
- Enter value 3: Textbox
**Action Buttons:**
- **Check:** Verifies the entered values.
- **Next:** Proceeds to the next step or problem.
#### Diagram Explanation:
The problem includes a vector \(\mathbf{y}\) represented in a column matrix format and a subspace \(W\) defined by the span of a single vector. The orthogonal projection of \(\mathbf{y}\) onto this subspace needs to be calculated. There are input boxes provided to allow the user to enter the components of the projected vector \(\hat{y}\).
### The Fundamental Theorem of Linear Algebra
The theorem below states which of the four fundamental subspaces associated with matrix \(A \in \mathbb{R}^{m \times n}\) are orthogonal.
**Theorem 7.5.6: The Fundamental Theorem of Linear Algebra**
---
**Feedback Section:** (Interactive)
---
This transcription includes a detailed description of the text and the structure of the problem. The vector \(\mathbf{y}\), the basis vector for subspace \(W\), the input boxes for the projection vector \(\hat{y}\), and the action buttons are clearly described.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f98d871-ff3c-46b6-bd04-f76052ab429d%2F750bf60d-2932-4194-9da8-cca0907d15d3%2F13bdb4r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Orthogonal Complements
### Section 7.5.2: Orthogonal Projections
#### Challenge Activity
**Problem Statement:**
Find the orthogonal projection \(\hat{y}\) of the vector \(\mathbf{y} = \begin{bmatrix} 6 \\ -2 \\ 2 \end{bmatrix}\) onto the subspace \(W = \text{Span} \left\{ \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix} \right\} \).
\[
\hat{y} =
\begin{bmatrix}
\text{Enter value 1} \\
\text{Enter value 2} \\
\text{Enter value 3}
\end{bmatrix}
\]
**Input Fields:**
- Enter value 1: Textbox
- Enter value 2: Textbox
- Enter value 3: Textbox
**Action Buttons:**
- **Check:** Verifies the entered values.
- **Next:** Proceeds to the next step or problem.
#### Diagram Explanation:
The problem includes a vector \(\mathbf{y}\) represented in a column matrix format and a subspace \(W\) defined by the span of a single vector. The orthogonal projection of \(\mathbf{y}\) onto this subspace needs to be calculated. There are input boxes provided to allow the user to enter the components of the projected vector \(\hat{y}\).
### The Fundamental Theorem of Linear Algebra
The theorem below states which of the four fundamental subspaces associated with matrix \(A \in \mathbb{R}^{m \times n}\) are orthogonal.
**Theorem 7.5.6: The Fundamental Theorem of Linear Algebra**
---
**Feedback Section:** (Interactive)
---
This transcription includes a detailed description of the text and the structure of the problem. The vector \(\mathbf{y}\), the basis vector for subspace \(W\), the input boxes for the projection vector \(\hat{y}\), and the action buttons are clearly described.
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