Q18E Consider the 4脳2 matrixA=110[11... [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: Q18E (page 412)

Consider the $4 脳 2$ matrix
$A = \frac{1}{10} [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 1 & - 1 & - 1 \\ 1 & - 1 & 1 & - 1 \\ 1 & - 1 & - 1 & 1 \end{matrix}] [\begin{matrix} 2 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} 3 & - 4 \\ 4 & 3 \end{matrix}]$
Use the result of Exercise 17 to find the least-squares solution of the linear system
$A \overset{鈫�}{x} = \overset{鈫�}{b}$ where $\overset{鈫�}{b} = [\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}]$
Work with paper and pencil.

Short Answer

Expert verified

${\overset{藱}{x}}^{*} = [\begin{matrix} - \frac{1}{10} \\ - \frac{16}{5} \end{matrix}]$

Step by step solution

To find the least-squares solution of the linear system

Given that,

$A = \frac{1}{10} [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 1 & - 1 & - 1 \\ 1 & - 1 & 1 & - 1 \\ 1 & - 1 & - 1 & 1 \end{matrix}] [\begin{matrix} 2 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} 3 & - 4 \\ 4 & 3 \end{matrix}]$

Therefore, we have $n = 4, m = 2, 蟽_{1} = 2, 蟽_{2} = 1$ . Also,

role="math" localid="1664872444698" $u_{1} = [\begin{matrix} \frac{1}{10} \\ \frac{1}{10} \\ \frac{1}{10} \\ \frac{1}{10} \end{matrix}], u_{2} = [\begin{matrix} \frac{1}{10} \\ \frac{1}{10} \\ - \frac{1}{10} \\ - \frac{1}{10} \end{matrix}], v_{1} = [\begin{matrix} 3 \\ - 4 \end{matrix}], v_{2} = [\begin{matrix} 4 \\ 3 \end{matrix}]$

Now the given linear system is:

$A x = [\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}]$

Thus $b = [\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}]$

To find the least-squares solution of the linear system

Therefore, using the result of Exercise 17 we get that the least squares solution of the given linear system is given by:

${\overset{藱}{x}}^{*} = \frac{b 路 \overset{藱}{u_{1}}}{蟽_{1}} \overset{藱}{v_{1}} + \frac{b 路 \overset{藱}{u_{2}}}{蟽_{2}} \overset{藱}{v_{2}} = \frac{\frac{1}{10} + \frac{2}{10} + \frac{3}{10} + \frac{4}{10}}{2} [\begin{matrix} 3 \\ - 4 \end{matrix}] + \frac{\frac{1}{10} + \frac{2}{10} - \frac{3}{10} - \frac{4}{10}}{1} [\begin{matrix} 4 \\ 3 \end{matrix}] = [\begin{matrix} \frac{3}{2} & - \frac{16}{10} \\ - 2 & - \frac{12}{10} \end{matrix}] = [\begin{matrix} \frac{- 1}{10} \\ \frac{- 16}{5} \end{matrix}]$

Final proof

Thus, the required least squares solution of the given linear system is ${\overset{藱}{x}}^{*} = [\begin{matrix} - \frac{1}{10} \\ \frac{- 16}{5} \end{matrix}]$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

To find the least-squares solution of the linear system

To find the least-squares solution of the linear system

Final proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Applied Mathematics

Mechanics Maths

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

91影视

Consider the 4脳2 matrixA=110[111111-1-11-11-11-1-11][20010000][3-443]Use the result of Exercise 17 to find the least-squares solution of the linear systemAx鈫�=b鈫�whereb鈫�=[1234]Work with paper and pencil.

Short Answer

Step by step solution

To find the least-squares solution of the linear system

To find the least-squares solution of the linear system

Final proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Applied Mathematics

Mechanics Maths

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.