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Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in R". If y is in a subspace W, then the orthogonal projection of y onto W is y itself. Choose the correct answer below. O A. The statement is true. For an orthogonal basis B= (u, u) of W, y and projwy can be written as linear combinations of vectors in B with equal weights. O B. The statement is true. If y is in W, then projwyy= -y, which is in the same spanning set as y. OC. The statement is false. If y is in W, then projwy =0. This means the statement is false unless y = 0. OD. The statement is false. If y is in W, then projwy is orthogonal to y and is in wt.

Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in R". If y is in a subspace W, then the orthogonal projection of y onto W is y itself. Choose the correct answer below. O A. The statement is true. For an orthogonal basis B= (u, u) of W, y and projwy can be written as linear combinations of vectors in B with equal weights. O B. The statement is true. If y is in W, then projwyy= -y, which is in the same spanning set as y. OC. The statement is false. If y is in W, then projwy =0. This means the statement is false unless y = 0. OD. The statement is false. If y is in W, then projwy is orthogonal to y and is in wt.

Advanced Engineering Mathematics
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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--- ### Problem Statement Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in \( \mathbb{R}^n \). **Statement:** If \( \textbf{y} \) is in a subspace \( W \), then the orthogonal projection of \( \textbf{y} \) onto \( W \) is \( \textbf{y} \) itself. **Question:** Choose the correct answer below. ### Answer Options **A.** The statement is true. For an orthogonal basis \( B = \{ \textbf{u}_1, \ldots, \textbf{u}_p \} \) of \( W \), and \( \text{proj}_W \textbf{y} \) can be written as linear combinations of vectors in \( B \) with equal weights. **B.** The statement is true. If \( \textbf{y} \) is in \( W \), then \( \text{proj}_W \textbf{y} = \textbf{y} \), which is in the same spanning set as \( \textbf{y} \). **C.** The statement is false. If \( \textbf{y} \) is in \( W \), then \( \text{proj}_W \textbf{y} = 0 \). This means the statement is false unless \( \textbf{y} = 0 \). **D.** The statement is false. If \( \textbf{y} \) is in \( W \), then \( \text{proj}_W \textbf{y} \) is orthogonal to \( \textbf{y} \) and is in \( W^\perp \). --- This section presents a true/false question based on linear algebra concepts, specifically dealing with orthogonal projections onto subspaces. Each answer choice provides a possible explanation for whether the given statement is true or false and why. The correct choice needs to be selected based on the understanding of projection properties and subspaces.
Transcribed Image Text:--- ### Problem Statement Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in \( \mathbb{R}^n \). **Statement:** If \( \textbf{y} \) is in a subspace \( W \), then the orthogonal projection of \( \textbf{y} \) onto \( W \) is \( \textbf{y} \) itself. **Question:** Choose the correct answer below. ### Answer Options **A.** The statement is true. For an orthogonal basis \( B = \{ \textbf{u}_1, \ldots, \textbf{u}_p \} \) of \( W \), and \( \text{proj}_W \textbf{y} \) can be written as linear combinations of vectors in \( B \) with equal weights. **B.** The statement is true. If \( \textbf{y} \) is in \( W \), then \( \text{proj}_W \textbf{y} = \textbf{y} \), which is in the same spanning set as \( \textbf{y} \). **C.** The statement is false. If \( \textbf{y} \) is in \( W \), then \( \text{proj}_W \textbf{y} = 0 \). This means the statement is false unless \( \textbf{y} = 0 \). **D.** The statement is false. If \( \textbf{y} \) is in \( W \), then \( \text{proj}_W \textbf{y} \) is orthogonal to \( \textbf{y} \) and is in \( W^\perp \). --- This section presents a true/false question based on linear algebra concepts, specifically dealing with orthogonal projections onto subspaces. Each answer choice provides a possible explanation for whether the given statement is true or false and why. The correct choice needs to be selected based on the understanding of projection properties and subspaces.
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