---### Problem StatementDetermine 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.

Answered: Image /qna-images/answer/e6a37732-96e1-4310-b187-0325d16f19ab.jpg

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

---

### 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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