Ton work ty 3.5] Consider the Fuclidean vector space IRS with the dot product. A subspace U c/R³ and X6IRS aregiven by. O U= Span [-1 2 NG 2 + 1 -3 1 -1 T -3 J124 ។ T -1 خان on + -1 -37 x = -9 -1 구 Fak Ч 2 1 „Determine the orthogonal projection Thu (x) ofx ento U b. Determine the distance dcx, U).
Ton work ty 3.5] Consider the Fuclidean vector space IRS with the dot product. A subspace U c/R³ and X6IRS aregiven by. O U= Span [-1 2 NG 2 + 1 -3 1 -1 T -3 J124 ។ T -1 خان on + -1 -37 x = -9 -1 구 Fak Ч 2 1 „Determine the orthogonal projection Thu (x) ofx ento U b. Determine the distance dcx, U).
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
3.5
![**3.5) Consider the Euclidean vector space \( \mathbb{R}^5 \) with the dot product.**
A subspace \( U \subset \mathbb{R}^5 \) and \( x \in \mathbb{R}^5 \) are given by:
\[ U = \text{span} \left\{ \begin{bmatrix} 0 \\ 2 \\ 2 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \\ 1 \\ -3 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 5 \\ 0 \\ 4 \end{bmatrix} \right\} \]
\[ x = \begin{bmatrix} -1 \\ -9 \\ -1 \\ 4 \\ 1 \end{bmatrix} \]
**a) Determine the orthogonal projection \( P_U(x) \) of \( x \) onto \( U \).**
**b) Determine the distance \( d(x, U) \).**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e25041d-7573-46df-b9d3-ec2dd7694c16%2F323f0cf4-05b0-402f-8c04-88f1ec6d54a2%2F5ttsfps_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**3.5) Consider the Euclidean vector space \( \mathbb{R}^5 \) with the dot product.**
A subspace \( U \subset \mathbb{R}^5 \) and \( x \in \mathbb{R}^5 \) are given by:
\[ U = \text{span} \left\{ \begin{bmatrix} 0 \\ 2 \\ 2 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \\ 1 \\ -3 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 5 \\ 0 \\ 4 \end{bmatrix} \right\} \]
\[ x = \begin{bmatrix} -1 \\ -9 \\ -1 \\ 4 \\ 1 \end{bmatrix} \]
**a) Determine the orthogonal projection \( P_U(x) \) of \( x \) onto \( U \).**
**b) Determine the distance \( d(x, U) \).**
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