Q15E Suppose a line L聽in R3聽contain... [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 2: Q15E (page 71)

Suppose a line Lin R³contains the unit vector
$\overset{鈫�}{u} = [\begin{array}{l} u_{1} \\ u_{2} \\ u_{3} \end{array}]$
Find the matrix Aof the linear transformation $T (\overset{鈫�}{x}) = r e f_{L} (\overset{鈫�}{x})$ . Give the entries of Ain terms localid="1664189516834" $u_{1}, u_{2}, u_{3}$ of the components of $\overset{鈫�}{u}$ .

Short Answer

Expert verified

The matrix $A = [\begin{matrix} 2 u_{1}^{2} 鈭� 1 & 2 u_{1} u_{2} & 2 u_{1} u_{3} \\ 2 u_{2} u_{1} & 2 u_{2}^{2} 鈭� 1 & 2 u_{2} u_{3} \\ 2 u_{3} u_{1} & 2 u_{3} u_{2} & 2 u_{3}^{2} 鈭� 1 \end{matrix}]$ , with $a_{i i} = 2 u_{i}^{2} 鈭� 1, 鈥� a_{i j} = 2 u_{i} u_{j} 鈥墂丑别苍鈥� i 鈮� j$ .

Step by step solution

Compute the projection of the vector

The projection of vector $\overset{鈫�}{x}$ onto $\overset{鈫�}{u}$ is,

${proj}_{L} (\overset{鈫�}{x}) = (\frac{\overset{鈫�}{x} 鈰� \overset{鈫�}{u}}{u u}) \overset{鈫�}{u}$

Here, $\overset{鈫�}{x} = [\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}] {鈭�}^{3}$ .

Thus, the projection is,

$\begin{matrix} {proj}_{L} (\overset{鈫�}{x}) = {proj}_{x} [\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}] \\ = ([\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}] 鈰� [\begin{array}{l} u_{1} \\ u_{2} \\ u_{3} \end{array}]) [\begin{array}{l} u_{1} \\ u_{2} \\ u_{3} \end{array}] \\ = [\begin{matrix} u_{1}^{2} & u_{1} u_{2} & u_{1} u_{3} \\ u_{2} u_{1} & u_{2}^{2} & u_{2} u_{3} \\ u_{3} u_{1} & u_{3} u_{2} & u_{3}^{2} \end{matrix}] [\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}] \end{matrix}$

Compute the reflection of the vector

The reflection of vector $\overset{鈫�}{x}$ is,

$r e f_{L} (\overset{鈫�}{x}) = 2 {proj}_{L} (\overset{鈫�}{x}) 鈭� (\overset{鈫�}{x})$

Here, $\overset{鈫�}{x} = [\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}] {鈭�}^{3}$

Thus, the reflection is,

$\begin{array}{l} r e f_{L} [\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}] = 2 [\begin{matrix} u_{1}^{2} & u_{1} u_{2} & u_{1} u_{3} \\ u_{2} u_{1} & u_{2}^{2} & u_{2} u_{3} \\ u_{3} u_{1} & u_{3} u_{2} & u_{3}^{2} \end{matrix}] [\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}] 鈭� [\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}] \\ r e f_{L} [\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}] = [\begin{matrix} 2 u_{1}^{2} 鈭� 1 & 2 u_{1} u_{2} & 2 u_{1} u_{3} \\ 2 u_{2} u_{1} & 2 u_{2}^{2} 鈭� 1 & 2 u_{2} u_{3} \\ 2 u_{3} u_{1} & 2 u_{3} u_{2} & 2 u_{3}^{2} 鈭� 1 \end{matrix}] [\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}] \end{array}$

Compute the matrix

As the matrix A is the linear transformation of $T (\overset{鈫�}{x}) = {ref}_{L} (\overset{鈫�}{x})$ ,

$A = [\begin{matrix} 2 u_{1}^{2} 鈭� 1 & 2 u_{1} u_{2} & 2 u_{1} u_{3} \\ 2 u_{2} u_{1} & 2 u_{2}^{2} 鈭� 1 & 2 u_{2} u_{3} \\ 2 u_{3} u_{1} & 2 u_{3} u_{2} & 2 u_{3}^{2} 鈭� 1 \end{matrix}]$

Hence, the matrix $A = [\begin{matrix} 2 u_{1}^{2} 鈭� 1 & 2 u_{1} u_{2} & 2 u_{1} u_{3} \\ 2 u_{2} u_{1} & 2 u_{2}^{2} 鈭� 1 & 2 u_{2} u_{3} \\ 2 u_{3} u_{1} & 2 u_{3} u_{2} & 2 u_{3}^{2} 鈭� 1 \end{matrix}]$ with $a_{i i} = 2 u_{i}^{2} 鈭� 1, 鈥� a_{i j} = 2 u_{i} u_{j} 鈥墂丑别苍鈥� i 鈮� j$ .

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91影视

Short Answer

Step by step solution

Compute the projection of the vector

Compute the reflection of the vector

Compute the matrix

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91影视

Suppose a line Lin R3contains the unit vectoru鈫�=[u1u2u3]Find the matrix Aof the linear transformation T(x鈫�)=refL(x鈫�). Give the entries of Ain terms localid="1664189516834" u1,u2,u3of the components of u鈫�.

Short Answer

Step by step solution

Compute the projection of the vector

Compute the reflection of the vector

Compute the matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Pure Maths

Mechanics Maths

Theoretical and Mathematical Physics

Geometry

Study anywhere. Anytime. Across all devices.