Q15E Using paper and pencil, find the... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q15E (page 224)

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.
15. $[\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}]$

Short Answer

Expert verified

The QR factorization of the matrices $[\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}] is [\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}] = \underset{Q}{\underset{铃�}{[\begin{matrix} 2 / 3 \\ 1 / 3 \\ - 2 / 3 \end{matrix}]}} \underset{R}{\underset{铃�}{3}}$

Step by step solution

QR factorization

Consider an $n 脳 m$ matrix Mwith linearly independent columns ${\overset{鈫�}{v}}_{1}, . . . . {\overset{鈫�}{v}}_{m}$ . Then there exists an $n 脳 m$ matrix Qwhose columns ${\overset{鈫�}{u}}_{1}, . . . . {\overset{鈫�}{u}}_{m}$ are orthonormal and an upper triangular matrix Rwith positive diagonal entries such that.

M=QR

Furthermore, $r_{11} = ||{\overset{鈫�}{v}}_{1}||, r_{j j} = ||{\overset{鈫�}{v}}_{j}^{鈯�}||, (For j = 2, . . . m) and r_{i j} = {\overset{鈫�}{u}}_{i} . {\overset{鈫�}{v}}_{j}$ .

According to above theorem

Let the given vector is, $M = [\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}], where {\overset{鈫�}{v}}_{1} = [\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}]$ , where .

Consider the terms below to find out the QR factorization.

$r_{11} = ||{\overset{鈫�}{v}}_{1}|| = \sqrt{2^{2} + 1^{2} + {(- 2)}^{2}} = \sqrt{4 + 1 + 4} = 3$

Now, find ${\overset{鈫�}{u}}_{1}$ .

${\overset{鈫�}{u}}_{1} = \frac{1}{r_{11}} {\overset{鈫�}{v}}_{1} = \frac{1}{3} [\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}] = [\begin{matrix} 2 / 3 \\ 1 / 3 \\ - 2 / 3 \end{matrix}]$

Thus, the value is $[\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}] = \underset{Q}{\underset{铃�}{[\begin{matrix} 2 / 3 \\ 1 / 3 \\ - 2 / 3 \end{matrix}]}} \underset{R}{\underset{铃�}{3}}$

Hence, the QR factorization for M is $[\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}] = \underset{Q}{\underset{铃�}{[\begin{matrix} 2 / 3 \\ 1 / 3 \\ - 2 / 3 \end{matrix}]}} \underset{R}{\underset{铃�}{3}}$ .

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91影视

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.
15. $[\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}]$

Short Answer

Step by step solution

QR factorization

According to above theorem

One App. One Place for Learning.

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91影视

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.15.[21-2]

Short Answer

Step by step solution

QR factorization

According to above theorem

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Discrete Mathematics

Applied Mathematics

Pure Maths

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.
15. $[\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}]$