Q44E TRUE OR FALSEIf V is a subspace ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q44E (page 264)

TRUE OR FALSE
If V is a subspace of $R^{n}$ and $\overset{鈫�}{x}$ is a vector space in $R^{n}$ , then the inequality $\overset{鈫�}{x} 路 (pr o_{V} \overset{鈫�}{x}) 猢� 0$ must hold.

Short Answer

Expert verified

The given statement is true.

Step by step solution

Definition of orthogonal projection

Vector $p r o j_{v} \overset{鈫�}{x}$ is orthogonal to vector $\overset{鈫�}{x} - p r o j_{v} \overset{鈫�}{x}$ .

Explanation of the statement

Consider the statement as follows.

鈥淚f V is a subspace of $R^{n}$ and $\overset{鈫�}{x}$ is a vector in $R^{n}$ , then the inequality $\overset{鈫�}{x} 路 p r o j_{v} \overset{鈫�}{x} 猢� 0$ 鈥�.

Since, $p r o j v \overset{鈫�}{x}$ is a orthogonal vector $\overset{鈫�}{x} - p r o j_{v} \overset{鈫�}{x}$ and as follows.

$p r o j v \overset{鈫�}{x} 路 (\overset{鈫�}{x} - p r o j_{v} \overset{鈫�}{x}) = 0$ 鈥︹赌�(1)

$\overset{鈫�}{x} 路 p r o j v \overset{鈫�}{x} = (\overset{鈫�}{x} + p r o j v \overset{鈫�}{x} - p r o j v \overset{鈫�}{x}) 路 (p r o j v \overset{鈫�}{x}) = (p r o j v \overset{鈫�}{x} + (\overset{鈫�}{x} + p r o j v \overset{鈫�}{x})) 路 (p r o j v \overset{鈫�}{x}) = (p r o j v \overset{鈫�}{x}) 路 (p r o j v \overset{鈫�}{x}) + (\overset{鈫�}{x} + p r o j v \overset{鈫�}{x}) 路 (p r o j v \overset{鈫�}{x}) \overset{鈫�}{x} 路 p r o j v \overset{鈫�}{x} = (p r o j v \overset{鈫�}{x}) 路 (p r o j v \overset{鈫�}{x}) + 0$

Simplify further as follows.

$\overset{鈫�}{x} 路 p r o j v \overset{鈫�}{x} = (p r o j v \overset{鈫�}{x}) 路 (p r o j v \overset{鈫�}{x}) + 0 = {||p r o j v \overset{鈫�}{x}||}^{2} 猢� 0$

Thus, the given condition 鈥淚f V is a subspace of $R^{n}$ and $\overset{鈫�}{x}$ is a vector in $R^{n}$ , then the inequality $\overset{鈫�}{x} 路 p r o j_{v} \overset{鈫�}{x} 猢� 0$ 鈥� is true.

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91影视

TRUE OR FALSE
If V is a subspace of $R^{n}$ and $\overset{鈫�}{x}$ is a vector space in $R^{n}$ , then the inequality $\overset{鈫�}{x} 路 (pr o_{V} \overset{鈫�}{x}) 猢� 0$ must hold.

Short Answer

Step by step solution

Definition of orthogonal projection

Explanation of the statement

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91影视

TRUE OR FALSEIf V is a subspace of Rnand x鈫�is a vector space in Rn, then the inequality x鈫�路(proVx鈫�)猢�0must hold.

Short Answer

Step by step solution

Definition of orthogonal projection

Explanation of the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Logic and Functions

Probability and Statistics

Study anywhere. Anytime. Across all devices.

TRUE OR FALSE
If V is a subspace of $R^{n}$ and $\overset{鈫�}{x}$ is a vector space in $R^{n}$ , then the inequality $\overset{鈫�}{x} 路 (pr o_{V} \overset{鈫�}{x}) 猢� 0$ must hold.