Problem 22. Find an orthonormal set 91,92,93 for which q1,92 span the column space of12-2 4A = =Which fundamental subspace contains q3? What is the least squares solution of Ax = b if b =[127]¹?

Given vector A=112-1-24 Find q1 as q1=c1c1. Therefore,…

Answered: Problem 22. Find an orthonormal set…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Problem 22. Find an orthonormal set 91,92,93 for which q1,92 span the column space of
1
2
-2 4
A = =
Which fundamental subspace contains q3? What is the least squares solution of Ax = b if b =
[127]¹?

Expert Solution

Step 1

Given vector $A = [\begin{matrix} 1 & 1 \\ 2 & - 1 \\ - 2 & 4 \end{matrix}]$

Find $q_{1}$ as $q_{1} = \frac{c_{1}}{||c_{1}||}$ .

Therefore,

$\begin{array}{rcl} q_{1} & = & \frac{1}{\sqrt{1^{2} + 2^{2} + {(- 2)}^{2}}} [\begin{matrix} 1 \\ 2 \\ - 2 \end{matrix}] \\ = & \frac{1}{\sqrt{1 + 4 + 4}} [\begin{matrix} 1 \\ 2 \\ - 2 \end{matrix}] \\ = & \frac{1}{\sqrt{9}} [\begin{matrix} 1 \\ 2 \\ - 2 \end{matrix}] \\ = & \frac{1}{3} [\begin{matrix} 1 \\ 2 \\ - 2 \end{matrix}] \\ = & [\begin{matrix} \frac{1}{3} \\ \frac{2}{3} \\ - \frac{2}{3} \end{matrix}] \end{array}$