Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q43E (page 1)

Consider a unit vector $\overset{鈯�}{u} in R^{3}$ . We define the matrices
$A = 2 \overset{鈯�}{u} \overset{鈯�}{u^{T}} - I_{3} and B - I_{3} - 2 \overset{鈯�}{u} \overset{鈯�}{u^{T}}$
Describe the linear transformations defined by these matrices geometrically.

Short Answer

Expert verified

A is the reflection of $\overset{鈯�}{x}$ about line L that is spanned by $\overset{鈯�}{u}$ .

B is the reflection of $\overset{鈯�}{x}$ about line V that is spanned by $\overset{鈯�}{u}$ .

Step by step solution

Reflection of x about line L.

Let the line L be spanned by a unit vector $\overset{鈯�}{u} in R^{3}$

By using the definition 2.2.2 the reflection of $\overset{鈯�}{x}$ about L is,

${ref}_{L} = 2 proj (\overset{鈯�}{x}) - \overset{鈯�}{x} = 2 (\overset{r}{x} . \overset{r}{u}) \overset{r}{u} - \overset{r}{x} = 2 \overset{r}{u} \overset{r}{u^{T} \overset{r}{x} - \overset{r}{x}} = (2 \overset{r}{u} \overset{r}{u^{T} - I_{3}}) \overset{r}{x} = A_{x}^{r}$

Reflection of x⊥ about line V.

Let Va plane with a normal vector $\overset{鈯�}{u}$

Then the reflection of $\overset{鈯�}{x}$ about line V.

$r e f_{v} = p r o j_{v} (\overset{鈯�}{X}) - p r o j_{L} \overset{鈯�}{X} = ((\overset{r}{X}) - p r o j_{L} \overset{r}{X}) - p r o j_{L} \overset{r}{X} = \overset{r}{X} - 2 p r o j_{L} \overset{r}{X} = \overset{r}{X} - 2 (\overset{r}{X} . \overset{r}{X}) \overset{r}{X} = \overset{r}{X} - 2 \overset{r}{u} \overset{r}{u^{T} \overset{r}{X}} = (I_{3} - 2 \overset{r}{u} \overset{r}{u^{T}}) \overset{r}{X} = B_{x}^{r}$

Hence, the answer is

A will be the reflection of $\overset{鈯�}{x}$ about line L that is spanned by $\overset{鈯�}{u}$ .

B will be the reflection of $\overset{鈯�}{x}$ about line V that is spanned by $\overset{鈯�}{u}$ .

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91影视

Consider a unit vector $\overset{鈯�}{u} in R^{3}$ . We define the matrices
$A = 2 \overset{鈯�}{u} \overset{鈯�}{u^{T}} - I_{3} and B - I_{3} - 2 \overset{鈯�}{u} \overset{鈯�}{u^{T}}$
Describe the linear transformations defined by these matrices geometrically.

Short Answer

Step by step solution

Reflection of x about line L.

Reflection of x⊥ about line V.

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91影视

Consider a unit vector u鈯�inR3. We define the matricesA=2u鈯�uT鈯�-I3andB-I3-2u鈯�uT鈯�Describe the linear transformations defined by these matrices geometrically.

Short Answer

Step by step solution

Reflection of x about line L.

Reflection of x⊥ about line V.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Statistics

Decision Maths

Theoretical and Mathematical Physics

Calculus

Study anywhere. Anytime. Across all devices.

Consider a unit vector $\overset{鈯�}{u} in R^{3}$ . We define the matrices
$A = 2 \overset{鈯�}{u} \overset{鈯�}{u^{T}} - I_{3} and B - I_{3} - 2 \overset{鈯�}{u} \overset{鈯�}{u^{T}}$
Describe the linear transformations defined by these matrices geometrically.