5 -2aud-3-4-4Une the 9ram schidt to dofenmicoarthopor mal bait for to vubrpaceof p Cwith thespenned byprocenAEudideau tuner pachut)and y

Given vectors are x→=-2-3-4, y→=-5-1-4 Let the orthogonal basis be u1→,u2→ by Gram-schmidt…

Answered: and -3 -4 Une the G procen to dotenmiro…

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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-2
aud
-3
-4
-4
Une the 9ram schidt to dofenmico
arthopor mal bait for to vubrpace
of p Cwith the
spenned by
procen
A
Eudideau tuner pachut)
and y

Expert Solution

Step 1

Given vectors are

$\vec{x} = [\begin{matrix} - 2 \\ - 3 \\ - 4 \end{matrix}], \vec{y} = [\begin{matrix} - 5 \\ - 1 \\ - 4 \end{matrix}]$

Let the orthogonal basis be $\{\vec{u_{1}}, \vec{u_{2}}\}$

by Gram-schmidt orthogonalization

$\vec{u_{1}} = [\begin{matrix} - 2 \\ - 3 \\ - 4 \end{matrix}]$

$\vec{u_{2}} = \vec{y} - \frac{<\vec{y}, \vec{u_{1}}>}{<\vec{u_{1}}, \vec{u_{1}}>} \vec{u_{1}} \Rightarrow \vec{u_{2}} = [\begin{matrix} - 5 \\ - 1 \\ - 4 \end{matrix}] - \frac{<[\begin{matrix} - 5 \\ - 1 \\ - 4 \end{matrix}], [\begin{matrix} - 2 \\ - 3 \\ - 4 \end{matrix}]>}{{(- 2)}^{2} + {(- 3)}^{2} + {(- 4)}^{2}} [\begin{matrix} - 2 \\ - 3 \\ - 4 \end{matrix}] \Rightarrow \vec{u_{2}} = [\begin{matrix} - 5 \\ - 1 \\ - 4 \end{matrix}] - \frac{10 + 3 + 16}{29} [\begin{matrix} - 2 \\ - 3 \\ - 4 \end{matrix}] \Rightarrow \vec{u_{2}} = [\begin{matrix} - 5 \\ - 1 \\ - 4 \end{matrix}] - \frac{29}{29} [\begin{matrix} - 2 \\ - 3 \\ - 4 \end{matrix}] \Rightarrow \vec{u_{2}} = [\begin{matrix} - 5 \\ - 1 \\ - 4 \end{matrix}] - [\begin{matrix} - 2 \\ - 3 \\ - 4 \end{matrix}] \Rightarrow \vec{u_{2}} = [\begin{matrix} - 3 \\ 2 \\ 0 \end{matrix}]$