**Educational Content on Orthogonality and Linear Independence****Exercise: Determining Truthfulness of a Statement****Problem:**"Determine whether the statement below is true or false. Justify the answer."**Statement:**"Not every orthogonal set in $ \mathbb{R}^n $ is linearly independent."**Multiple Choice Options:**A. **The statement is true.** Orthogonal sets with fewer than $ n $ vectors in $ \mathbb{R}^n $ are not linearly independent.B. **The statement is false.** Every orthogonal set of nonzero vectors is linearly independent and zero vectors cannot exist in orthogonal sets.C. **The statement is false.** Orthogonal sets must be linearly independent in order to be orthogonal.D. **The statement is true.** Every orthogonal set of nonzero vectors is linearly independent, but not every orthogonal set is linearly independent.### Explanation:The problem requires determining the truthfulness of the statement concerning orthogonal sets in $ \mathbb{R}^n $ and their linear independence properties. Let’s break down the concept:1. **Orthogonal Set:** A set of vectors is orthogonal if every pair of distinct vectors in the set is orthogonal, meaning their dot product is zero.2. **Linear Independence:** A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others.#### Analysis of Answer Choices:- **Option A:** Suggests that orthogonal sets with fewer than $ n $ vectors in $ \mathbb{R}^n $ are not linearly independent. This is misleading because orthogonality and linear independence are properties that do not directly relate to the number of vectors for fewer than $ n $ (as long as none are zero vectors, they are independent).- **Option B:** Asserts that all orthogonal sets of nonzero vectors are linearly independent and that zero vectors cannot exist in such sets. This is true but does not address the original statement, which considers orthogonal sets in general.- **Option C:** Claims that orthogonal sets must be linearly independent to be orthogonal. This is incorrect. Orthogonality doesn't require linear independence outright if zero vectors are present.- **Option D:** Accurate evaluation; it states that while every orthogonal set of nonzero vectors

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Determine whether the statement below is true or false. Justify the answer. Not every orthogonal sot in R" is linearly independent. Choose the correct answer below. OA. The statement is true. Orthogonal sets with fewer than n vectors in R" are not linearly independent. B. The statement is false. Every orthogonal set of nonzero vectors is linearly independent and zero vectors cannot exist in orthogonal sets. OC. The statement is false. Orthogonal sets must be linearly independent in order to be orthogonal. O D. The statement is true. Every orthogonal set of nonzero vectors is linearly independent, but not every orthogonal set is linearly independent.

Determine whether the statement below is true or false. Justify the answer. Not every orthogonal sot in R" is linearly independent. Choose the correct answer below. OA. The statement is true. Orthogonal sets with fewer than n vectors in R" are not linearly independent. B. The statement is false. Every orthogonal set of nonzero vectors is linearly independent and zero vectors cannot exist in orthogonal sets. OC. The statement is false. Orthogonal sets must be linearly independent in order to be orthogonal. O D. The statement is true. Every orthogonal set of nonzero vectors is linearly independent, but not every orthogonal set is linearly independent.

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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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

**Educational Content on Orthogonality and Linear Independence**

**Exercise: Determining Truthfulness of a Statement**

**Problem:**
"Determine whether the statement below is true or false. Justify the answer."

**Statement:**
"Not every orthogonal set in $ \mathbb{R}^n $ is linearly independent."

**Multiple Choice Options:**

A. **The statement is true.** Orthogonal sets with fewer than $ n $ vectors in $ \mathbb{R}^n $ are not linearly independent.

B. **The statement is false.** Every orthogonal set of nonzero vectors is linearly independent and zero vectors cannot exist in orthogonal sets.

C. **The statement is false.** Orthogonal sets must be linearly independent in order to be orthogonal.

D. **The statement is true.** Every orthogonal set of nonzero vectors is linearly independent, but not every orthogonal set is linearly independent.

### Explanation:

The problem requires determining the truthfulness of the statement concerning orthogonal sets in $ \mathbb{R}^n $ and their linear independence properties. Let’s break down the concept:

1. **Orthogonal Set:** A set of vectors is orthogonal if every pair of distinct vectors in the set is orthogonal, meaning their dot product is zero.
2. **Linear Independence:** A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others.

#### Analysis of Answer Choices:

- **Option A:** Suggests that orthogonal sets with fewer than $ n $ vectors in $ \mathbb{R}^n $ are not linearly independent. This is misleading because orthogonality and linear independence are properties that do not directly relate to the number of vectors for fewer than $ n $ (as long as none are zero vectors, they are independent).

- **Option B:** Asserts that all orthogonal sets of nonzero vectors are linearly independent and that zero vectors cannot exist in such sets. This is true but does not address the original statement, which considers orthogonal sets in general.

- **Option C:** Claims that orthogonal sets must be linearly independent to be orthogonal. This is incorrect. Orthogonality doesn't require linear independence outright if zero vectors are present.

- **Option D:** Accurate evaluation; it states that while every orthogonal set of nonzero vectors

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