Q30E Find a decomposition聽A=蟽1u鈫�1... [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 8: Q30E (page 413)

Find a decomposition $A = 蟽_{1} {\overset{鈫�}{u}}_{1} {\overset{鈫�}{v}}_{1}^{T} + 蟽_{2} {\overset{鈫�}{u}}_{2} {\overset{鈫�}{v}}_{2}^{T}$
$f o r A = [\begin{matrix} 6 & 2 \\ - 7 & 6 \end{matrix}]$ See Exercise 29 and Example 2.

Short Answer

Expert verified

$A = 10 [\begin{matrix} \frac{1}{\sqrt{5}} \\ - \frac{2}{\sqrt{5}} \end{matrix}] [\begin{matrix} \frac{2}{\sqrt{5}} & - \frac{1}{\sqrt{5}} \end{matrix}] + 5 [\begin{matrix} \frac{2}{\sqrt{5}} \\ \frac{1}{\sqrt{5}} \end{matrix}] [\begin{matrix} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \end{matrix}]$

Step by step solution

To finda decomposition

Given that, $A = [\begin{matrix} 6 & 2 \\ - 7 & 6 \end{matrix}]$

Now from Example 2 we have $v_{1} = \frac{1}{\sqrt{5}} [\begin{matrix} 2 \\ - 1 \end{matrix}]$

and $v_{2} = \frac{1}{\sqrt{5}} [\begin{matrix} 1 \\ 2 \end{matrix}]$

To find a decomposition

Also, $蟽_{1} = 10 a n d 蟽_{2} = 5$ Then

role="math" localid="1660642757479" $u_{1} = \frac{1}{蟽_{1}} A v_{1} = \frac{1}{10} [\begin{matrix} 6 & 2 \\ - 7 & 6 \end{matrix}] [\begin{matrix} \frac{2}{\sqrt{5}} \\ - \frac{1}{\sqrt{5}} \end{matrix}] = [\begin{matrix} \frac{1}{\sqrt{5}} \\ - \frac{2}{\sqrt{5}} \end{matrix}] {}_{2}= \frac{1}{蟽_{2}} A v_{2} = \frac{1}{5} [\begin{matrix} 6 & 2 \\ - 7 & 6 \end{matrix}] [\begin{matrix} \frac{1}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} \end{matrix}] = [\begin{matrix} \frac{2}{\sqrt{5}} \\ \frac{1}{\sqrt{5}} \end{matrix}]$

Thus the required decomposition is

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91影视

Find a decomposition $A = 蟽_{1} {\overset{鈫�}{u}}_{1} {\overset{鈫�}{v}}_{1}^{T} + 蟽_{2} {\overset{鈫�}{u}}_{2} {\overset{鈫�}{v}}_{2}^{T}$
$f o r A = [\begin{matrix} 6 & 2 \\ - 7 & 6 \end{matrix}]$ See Exercise 29 and Example 2.

Short Answer

Step by step solution

To finda decomposition

To find a decomposition

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91影视

Find a decompositionA=蟽1u鈫�1v鈫�1T+蟽2u鈫�2v鈫�2TforA=[62-76]See Exercise 29 and Example 2.

Short Answer

Step by step solution

To finda decomposition

To find a decomposition

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Decision Maths

Discrete Mathematics

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Find a decomposition $A = 蟽_{1} {\overset{鈫�}{u}}_{1} {\overset{鈫�}{v}}_{1}^{T} + 蟽_{2} {\overset{鈫�}{u}}_{2} {\overset{鈫�}{v}}_{2}^{T}$
$f o r A = [\begin{matrix} 6 & 2 \\ - 7 & 6 \end{matrix}]$ See Exercise 29 and Example 2.