Let W be the subspace spanned by u₁ and u₂, and write y as the sum of a vector in W and a vector orthogonal to W. 000 3, U₁ y = -2 5 The sum is y = y + z, where y= (Simplify your answers.) -3 - 1 is in W and Z= is orthogonal to W.
Let W be the subspace spanned by u₁ and u₂, and write y as the sum of a vector in W and a vector orthogonal to W. 000 3, U₁ y = -2 5 The sum is y = y + z, where y= (Simplify your answers.) -3 - 1 is in W and Z= is orthogonal to W.
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
![**Problem Statement:**
Let \( W \) be the subspace spanned by \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \), and write \( \mathbf{y} \) as the sum of a vector in \( W \) and a vector orthogonal to \( W \).
\[
\mathbf{y} = \begin{bmatrix} -2 \\ 3 \\ 5 \end{bmatrix}
\]
\[
\mathbf{u}_1 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{u}_2 = \begin{bmatrix} -3 \\ 4 \\ -1 \end{bmatrix}
\]
---
**Equation:**
The sum is \( \mathbf{y} = \hat{\mathbf{y}} + \mathbf{z} \), where \(\hat{\mathbf{y}} = \begin{bmatrix} \text{box} \\ \text{box} \\ \text{box} \end{bmatrix} \) is in \( W \) and \(\mathbf{z} = \begin{bmatrix} \text{box} \\ \text{box} \\ \text{box} \end{bmatrix} \) is orthogonal to \( W \).
*(Simplify your answers.)*
---
This problem involves expressing a given vector, \( \mathbf{y} \), as the sum of two vectors: one that lies within a specified subspace \( W \), and another that is orthogonal to it. The subspace \( W \) is defined by the vectors \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \). The challenge is to simplify and find the components \(\hat{\mathbf{y}}\) and \(\mathbf{z}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F95ac1400-18fa-42f2-b14c-acf34ce9de79%2Fcc7e620b-ad47-47bd-ab2a-020d146923ae%2Fk0mjjjo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( W \) be the subspace spanned by \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \), and write \( \mathbf{y} \) as the sum of a vector in \( W \) and a vector orthogonal to \( W \).
\[
\mathbf{y} = \begin{bmatrix} -2 \\ 3 \\ 5 \end{bmatrix}
\]
\[
\mathbf{u}_1 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{u}_2 = \begin{bmatrix} -3 \\ 4 \\ -1 \end{bmatrix}
\]
---
**Equation:**
The sum is \( \mathbf{y} = \hat{\mathbf{y}} + \mathbf{z} \), where \(\hat{\mathbf{y}} = \begin{bmatrix} \text{box} \\ \text{box} \\ \text{box} \end{bmatrix} \) is in \( W \) and \(\mathbf{z} = \begin{bmatrix} \text{box} \\ \text{box} \\ \text{box} \end{bmatrix} \) is orthogonal to \( W \).
*(Simplify your answers.)*
---
This problem involves expressing a given vector, \( \mathbf{y} \), as the sum of two vectors: one that lies within a specified subspace \( W \), and another that is orthogonal to it. The subspace \( W \) is defined by the vectors \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \). The challenge is to simplify and find the components \(\hat{\mathbf{y}}\) and \(\mathbf{z}\).
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