Q30E Consider a subspace V聽of Rn聽an... [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: Q30E (page 217)

Consider a subspace Vof $R^{n}$ and a vector $\overset{鈫�}{x}$ in $R^{n}$ . Let $\overset{鈫�}{y} = p r o j_{V} \overset{鈫�}{x}$ . What is the relationship between the following quantities?
$| | \overset{鈫�}{y} | |^{2}$ and $\overset{鈫�}{y} . \overset{鈫�}{x}$

Short Answer

Expert verified

The relationship is $| | \overset{鈫�}{y} | |^{2} = \overset{鈫�}{y} 路 \overset{鈫�}{x}$ ,

Step by step solution

Step by step solution Step 1: Draw the diagram

Draw the diagram which represent the given condition as:

Determine the relationship

Let us first recall the geometric meaning of $\overset{鈫�}{y} = pro j_{V} \overset{鈫�}{x}$ . From the above diagram, it is clear that $\overset{鈫�}{y} - \overset{鈫�}{x}$ is perpendicular to $\overset{鈫�}{y}$ and it is denoted by $x^{| |} = \overset{鈫�}{x} - {\overset{鈫�}{x}}^{鈯�}$ .

Thus, we have:

$\overset{鈫�}{y} \overset{鈫�}{x} 鈯� \overset{鈫�}{y} 鈬� (\overset{鈫�}{y} - \overset{鈫�}{x}) 路 \overset{鈫�}{y} = 0 鈬� \overset{鈫�}{y} . \overset{鈫�}{y} - \overset{鈫�}{x} 路 \overset{鈫�}{y} = 0 鈬� | | \overset{鈫�}{y} | |^{2} - \overset{鈫�}{x} 路 \overset{鈫�}{y} = 0 鈬� | | \overset{鈫�}{y} | |^{2} = \overset{鈫�}{x} 路 \overset{鈫�}{y} = \overset{鈫�}{y} 路 \overset{鈫�}{x}$

Hence, Relation between the $| | \overset{鈫�}{y} | |^{2}$ and $\overset{鈫�}{y} 路 \overset{鈫�}{x}$ is $| | \overset{鈫�}{y} | |^{2} = \overset{鈫�}{y} 路 \overset{鈫�}{x}$ .

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91影视

Consider a subspace Vof $R^{n}$ and a vector $\overset{鈫�}{x}$ in $R^{n}$ . Let $\overset{鈫�}{y} = p r o j_{V} \overset{鈫�}{x}$ . What is the relationship between the following quantities?
$| | \overset{鈫�}{y} | |^{2}$ and $\overset{鈫�}{y} . \overset{鈫�}{x}$

Short Answer

Step by step solution

Step by step solution Step 1: Draw the diagram

Determine the relationship

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91影视

Consider a subspace Vof Rnand a vector x鈫� inRn. Let y鈫�=projVx鈫�. What is the relationship between the following quantities? ||y鈫�||2and y鈫�.x鈫�

Short Answer

Step by step solution

Step by step solution Step 1: Draw the diagram

Determine the relationship

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Geometry

Statistics

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Consider a subspace Vof $R^{n}$ and a vector $\overset{鈫�}{x}$ in $R^{n}$ . Let $\overset{鈫�}{y} = p r o j_{V} \overset{鈫�}{x}$ . What is the relationship between the following quantities?
$| | \overset{鈫�}{y} | |^{2}$ and $\overset{鈫�}{y} . \overset{鈫�}{x}$