11 please show work 11. Consider a set of linear independent vectors{ ,x,; } in R". Let a vectorv,=x, and a vector v, be computed aspfollows:(Eq A) v, = R,-E, , for 2spsi.X.:Vk= X,р2

Answered: • v, = X,- Lie = 5,-E k V, for 2

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

11 please show work

11. Consider a set of linear independent vectors
{ ,x,; } in R". Let a vector
v,=x, and a vector v, be computed as
p
follows:
(Eq A) v, = R,-E, , for 2spsi.
X.:V
k
= X,
р
2<p<i.
k 9
k=1
Vk
k
Show that voV,=0_for any 1<Q,T<i &
Q#T.

Expert Solution

Step 1

(Note: There is typo error in question while defining v_{p}. The summation should run k=1 to p-1, because by that time v_{p} is not defined)

Given the set of vectors $\{\vec{x_{1}}, \vec{x_{2}}, \dots, \vec{x_{l}}\}$ in $ℝ^{n}$ are linearly independent.

Define:

$\vec{v_{1}} = \vec{x_{1}} \vec{v_{p}} = \vec{x_{p}} - \sum_{k = 1}^{p - 1} (\frac{\vec{x_{p}} . \vec{v_{k}}}{\vec{v_{k}} . \vec{v_{k}}}) \vec{v_{k}} for 2 \leq p \leq l$

Clearly:

$\begin{array}{rcl} \vec{v_{1}} . \vec{v_{2}} & = & \vec{x_{1}} . (\vec{x_{2}} - (\frac{\vec{x_{2}} . \vec{v_{1}}}{\vec{v_{1}} . \vec{v_{1}}}) \vec{v_{1}}) \\ = & \vec{x_{1}} . \vec{x_{2}} - (\frac{\vec{x_{2}} . \vec{v_{1}}}{\vec{v_{1}} . \vec{v_{1}}}) \vec{v_{1}} . \vec{x_{1}} \\ = & \vec{v_{1}} . \vec{x_{2}} - (\frac{\vec{x_{2}} . \vec{v_{1}}}{\vec{x_{1}} . \vec{x_{1}}}) \vec{x_{1}} . \vec{x_{1}} \\ = & 0 \end{array}$