Find the closest point to y in the subspace W spanned by V₁ and V₂ 2 4 -6 2 3 3 2 y = a. O O a O с 37 11 36 11 11 36 11 b. 38 11 36 11 11 36 11 0 38 11 36 11 11 36 11 d. 38 11 36 11 11 37 11
Find the closest point to y in the subspace W spanned by V₁ and V₂ 2 4 -6 2 3 3 2 y = a. O O a O с 37 11 36 11 11 36 11 b. 38 11 36 11 11 36 11 0 38 11 36 11 11 36 11 d. 38 11 36 11 11 37 11
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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![## Finding the Closest Point in a Subspace
Given the problem statement:
Find the closest point to \( y \) in the subspace \( W \) spanned by \( v_1 \) and \( v_2 \).
The vectors and subspace are defined as follows:
\[ y = \begin{bmatrix} 2 \\ -6 \\ 3 \\ 2 \end{bmatrix} , \quad v_1 = \begin{bmatrix} 4 \\ 2 \\ 3 \\ -2 \end{bmatrix} , \quad v_2 = \begin{bmatrix} 1 \\ -1 \\ 0 \\ 1 \end{bmatrix}. \]
You need to choose the correct closest point from the following options:
a.
\[ \begin{bmatrix} \frac{37}{11} \\ -\frac{36}{11} \\ \frac{1}{11} \\ \frac{36}{11} \end{bmatrix} \]
b.
\[ \begin{bmatrix} \frac{38}{11} \\ -\frac{36}{11} \\ \frac{2}{11} \\ \frac{36}{11} \end{bmatrix} \]
c.
\[ \begin{bmatrix} \frac{38}{11} \\ -\frac{36}{11} \\ \frac{1}{11} \\ \frac{36}{11} \end{bmatrix} \]
d.
\[ \begin{bmatrix} \frac{38}{11} \\ -\frac{36}{11} \\ \frac{1}{11} \\ \frac{37}{11} \end{bmatrix} \]
### Answer Choices
- \(\mathbf{a}\)
- \(\mathbf{b}\)
- \(\mathbf{c}\)
- \(\mathbf{d}\)
### Instructions for Students
To find the closest point, you need to understand the concept of projecting vectors onto subspaces. This involves using the Gram-Schmidt process or directly applying the projection formulas for the given subspace.
Reflect on the properties of orthogonal projections and the calculations involved to determine which of the provided vectors is the closest in terms of Euclidean distance to the vector \( y \) in the subspace \( W \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb7f7b58d-6888-4348-bb09-a3d5baf90082%2Fac708c65-6cb3-4b17-9fae-73bb34622301%2Ftduoihi_processed.png&w=3840&q=75)
Transcribed Image Text:## Finding the Closest Point in a Subspace
Given the problem statement:
Find the closest point to \( y \) in the subspace \( W \) spanned by \( v_1 \) and \( v_2 \).
The vectors and subspace are defined as follows:
\[ y = \begin{bmatrix} 2 \\ -6 \\ 3 \\ 2 \end{bmatrix} , \quad v_1 = \begin{bmatrix} 4 \\ 2 \\ 3 \\ -2 \end{bmatrix} , \quad v_2 = \begin{bmatrix} 1 \\ -1 \\ 0 \\ 1 \end{bmatrix}. \]
You need to choose the correct closest point from the following options:
a.
\[ \begin{bmatrix} \frac{37}{11} \\ -\frac{36}{11} \\ \frac{1}{11} \\ \frac{36}{11} \end{bmatrix} \]
b.
\[ \begin{bmatrix} \frac{38}{11} \\ -\frac{36}{11} \\ \frac{2}{11} \\ \frac{36}{11} \end{bmatrix} \]
c.
\[ \begin{bmatrix} \frac{38}{11} \\ -\frac{36}{11} \\ \frac{1}{11} \\ \frac{36}{11} \end{bmatrix} \]
d.
\[ \begin{bmatrix} \frac{38}{11} \\ -\frac{36}{11} \\ \frac{1}{11} \\ \frac{37}{11} \end{bmatrix} \]
### Answer Choices
- \(\mathbf{a}\)
- \(\mathbf{b}\)
- \(\mathbf{c}\)
- \(\mathbf{d}\)
### Instructions for Students
To find the closest point, you need to understand the concept of projecting vectors onto subspaces. This involves using the Gram-Schmidt process or directly applying the projection formulas for the given subspace.
Reflect on the properties of orthogonal projections and the calculations involved to determine which of the provided vectors is the closest in terms of Euclidean distance to the vector \( y \) in the subspace \( W \).
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