3. Consider the Euclideam rector space IR³ with the dot product. A sobscape Subspace U≤IR³ and XEIRS are given by : -3 O U = Span [ O 2 -3 1 -1 2 -1 -3 5 O 7 x= 191. -9 -1 4 aj Determine the orthogonal projection πTUCK) of x onto U b) Determine the distance d(x, U)
3. Consider the Euclideam rector space IR³ with the dot product. A sobscape Subspace U≤IR³ and XEIRS are given by : -3 O U = Span [ O 2 -3 1 -1 2 -1 -3 5 O 7 x= 191. -9 -1 4 aj Determine the orthogonal projection πTUCK) of x onto U b) Determine the distance d(x, U)
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
3.5
![**Exercise on Euclidean Vector Spaces**
**Problem Statement:**
Consider the Euclidean vector space \(\mathbb{R}^5\) with the dot product. A subspace \(U \subset \mathbb{R}^5\) and a vector \(x \in \mathbb{R}^5\) are given by:
\[ U = \text{Span} \left\{ \begin{bmatrix} -1 \\ 2 \\ 0 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} -3 \\ 4 \\ 1 \\ 2 \\ 7 \end{bmatrix} \right\} \]
\[ x = \begin{bmatrix} -1 \\ -9 \\ -1 \\ 4 \\ 1 \end{bmatrix} \]
**Tasks:**
a) Determine the orthogonal projection \(\Pi_U(x)\) of \(x\) onto \(U\).
b) Determine the distance \(d(x, U)\).
**Instructions:**
- To solve task (a), use the formula for orthogonal projection of a vector onto a subspace spanned by other vectors.
- For task (b), calculate the distance between the vector \(x\) and the subspace \(U\) using the result of the orthogonal projection.
**Note:**
Understanding orthogonal projections and distances in vector spaces is essential for applications in areas such as data science, engineering, and physics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e25041d-7573-46df-b9d3-ec2dd7694c16%2Ffb13147e-bf9b-4f55-a423-1a4b80b015ed%2Frk973td_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise on Euclidean Vector Spaces**
**Problem Statement:**
Consider the Euclidean vector space \(\mathbb{R}^5\) with the dot product. A subspace \(U \subset \mathbb{R}^5\) and a vector \(x \in \mathbb{R}^5\) are given by:
\[ U = \text{Span} \left\{ \begin{bmatrix} -1 \\ 2 \\ 0 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} -3 \\ 4 \\ 1 \\ 2 \\ 7 \end{bmatrix} \right\} \]
\[ x = \begin{bmatrix} -1 \\ -9 \\ -1 \\ 4 \\ 1 \end{bmatrix} \]
**Tasks:**
a) Determine the orthogonal projection \(\Pi_U(x)\) of \(x\) onto \(U\).
b) Determine the distance \(d(x, U)\).
**Instructions:**
- To solve task (a), use the formula for orthogonal projection of a vector onto a subspace spanned by other vectors.
- For task (b), calculate the distance between the vector \(x\) and the subspace \(U\) using the result of the orthogonal projection.
**Note:**
Understanding orthogonal projections and distances in vector spaces is essential for applications in areas such as data science, engineering, and physics.
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