12 please show work equation a is in second picture Select any 4 vectors in ℜⁿ of your choice. Calculate **$\vec{v}_p$**'s, applying Eq A on these vectors. The image presents a mathematical expression labeled as Equation A, which is part of a broader context indicated by the word "follows:"\[ \vec{v}_p = \vec{x}_p - \sum_{k=1}^{p} \left( \frac{\vec{x}_p \cdot \vec{v}_k}{\vec{v}_k \cdot \vec{v}_k} \right) \vec{v}_k, \quad \text{for} \quad 2 \leq p \leq i.\]Additionally, there is an instruction to "Show that $ \vec{v}_Q \cdot \vec{v}_T = 0 $ for any $ 1 \leq Q, T \leq i $ and $ Q \neq T $."This suggests a process of orthogonalization, often related to the Gram-Schmidt procedure, where vectors $ \vec{v}_p $ are being orthogonally projected from input vectors $ \vec{x}_p $. The aim is to demonstrate that any two vectors $ \vec{v}_Q $ and $ \vec{v}_T $ are orthogonal if they are distinct (i.e., $ Q \neq T $).

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

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

equation a is in second picture

Select any 4 vectors in ℜⁿ of your choice. Calculate **$\vec{v}_p$**'s, applying Eq A on these vectors.

The image presents a mathematical expression labeled as Equation A, which is part of a broader context indicated by the word "follows:"

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

Additionally, there is an instruction to "Show that $ \vec{v}_Q \cdot \vec{v}_T = 0 $ for any $ 1 \leq Q, T \leq i $ and $ Q \neq T $."

This suggests a process of orthogonalization, often related to the Gram-Schmidt procedure, where vectors $ \vec{v}_p $ are being orthogonally projected from input vectors $ \vec{x}_p $. The aim is to demonstrate that any two vectors $ \vec{v}_Q $ and $ \vec{v}_T $ are orthogonal if they are distinct (i.e., $ Q \neq T $).

Expert Solution

Step 1

There is a misprint in the question given. In the equation (A) given in the question, the summation must run for k=1 to p-1 but it is given k=1 to p. Note if we consider the statement of the equation to be true then while computing v_p we are using the value of v_p in the summation which is not been known yet. The process in equation (A) in the question is known as Gram-schmidt orthogonalization process. Any standard textbook of linear algebra will lead to the correct formula.We will use here the correct formula to evalutate it.

Solution:

Let us consider the vector space $ℝ^{n}, n \geq 1$ .

Let us now consider 4 vectors from $ℝ^{n}$ which are $\vec{x_{1}}, \vec{x_{2}}, \vec{x_{3}}, \vec{x_{4}}$ .

Therefore we have to obtain an orthogonal set ${\vec{v_{1}}, \vec{v_{2}}, \vec{v_{3}}, \vec{v_{4}}}$ from $A = {\vec{x_{1}}, \vec{x_{2}}, \vec{x_{3}}, \vec{x_{4}}}$ by Gram-schmidt process, that is by using the equation (A) below.

Let us rewrite the equation (A):

$\vec{v_{p}} = \vec{x_{p}} - \sum_{k = 1}^{p - 1} (\frac{\vec{x_{p}} . \vec{v_{k}}}{\vec{v_{k} .} \vec{v_{k}}}) \vec{v_{k}}, f o r 2 \leq p \leq 4 (H e r e i = 4 a s w e h a v e t o c o n s i d e r 4 v e c t o r s o f ℝ^{n})$