12 please show work 12. Select any 4 vectors in $\mathbb{R}^n$ of your choice. Calculate $\vec{v}_p$'s, applying Eq A on these vectors. The equation shown, labeled as (Eq A), is as follows:\[\vec{v}_p = \vec{x}_p - \sum_{k=1}^{p-1} \left( \frac{\vec{x}_p \cdot \vec{v}_k}{\vec{v}_k \cdot \vec{v}_k} \right) \vec{v}_k, \quad \text{for } 2 \leq p \leq i.\]The task is to show that \[\vec{v}_Q \cdot \vec{v}_T = 0\]for any $ 1 \leq Q, T \leq i $ and $ Q \neq T $.This equation demonstrates the process of obtaining an orthogonal vector $\vec{v}_p$ by subtracting the projections of $\vec{x}_p$ onto previously determined orthogonal vectors $\vec{v}_k$. The subsequent task requires proving the orthogonality of vectors $\vec{v}_Q$ and $\vec{v}_T$ for distinct indices $Q$ and $T$.

Consider the four vectors on ℝn as follows: x1→=10000⋮0,x2→=11000⋮0,x3→=11100⋮0,x4→=11110⋮0 that is…

Answered: Select any 4 vectors in R" of your…

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

12 please show work

12. Select any 4 vectors in $\mathbb{R}^n$ of your choice. Calculate $\vec{v}_p$'s, applying Eq A on these vectors.

The equation shown, labeled as (Eq A), is as follows:

\[
\vec{v}_p = \vec{x}_p - \sum_{k=1}^{p-1} \left( \frac{\vec{x}_p \cdot \vec{v}_k}{\vec{v}_k \cdot \vec{v}_k} \right) \vec{v}_k, \quad \text{for } 2 \leq p \leq i.
\]

The task is to show that

\[
\vec{v}_Q \cdot \vec{v}_T = 0
\]

for any $ 1 \leq Q, T \leq i $ and $ Q \neq T $.

This equation demonstrates the process of obtaining an orthogonal vector $\vec{v}_p$ by subtracting the projections of $\vec{x}_p$ onto previously determined orthogonal vectors $\vec{v}_k$. The subsequent task requires proving the orthogonality of vectors $\vec{v}_Q$ and $\vec{v}_T$ for distinct indices $Q$ and $T$.

Expert Solution

Step 1

Consider the four vectors on $ℝ^{n}$ as follows:

$\vec{x_{1}} = [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ ⋮ \\ 0 \end{matrix}], \vec{x_{2}} = [\begin{matrix} 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ ⋮ \\ 0 \end{matrix}], \vec{x_{3}} = [\begin{matrix} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ ⋮ \\ 0 \end{matrix}], \vec{x_{4}} = [\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \\ 0 \\ ⋮ \\ 0 \end{matrix}]$