Consider a set of linear independent vectors { x,,x2; } in R". Let a vector v,=x, and a vector v, be computed as ... follows: p-1 | Xp •V k v, = x,-E Vk, k=1\ V•V, for 2
Consider a set of linear independent vectors { x,,x2; } in R". Let a vector v,=x, and a vector v, be computed as ... follows: p-1 | Xp •V k v, = x,-E Vk, k=1\ V•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
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Question
11 please show work
![**Orthogonal Vector Set Calculation**
**Problem Statement:**
Consider a set of linearly independent vectors \(\{\vec{x}_1, \vec{x}_2, \ldots, \vec{x}_i\}\) in \(\mathbb{R}^n\). Let a vector \(\vec{v}_1 = \vec{x}_1\) and a vector \(\vec{v}_p\) be computed as follows:
**Equation A:**
\[
\vec{v}_p = \vec{x}_p - \sum_{k=1}^{p-1} \left( \frac{\vec{x}_p \cdot \vec{v}_k}{\vec{v}_k \cdot \vec{v}_k} \right) \vec{v}_k, \quad \text{for} \quad 2 \leq p \leq i.
\]
**Objective:**
Show that \(\vec{v}_Q \cdot \vec{v}_T = 0\) for any \(1 \leq Q, T \leq i\) and \(Q \neq T\).
**Explanation:**
This problem involves creating an orthogonal set of vectors from a linearly independent set using the Gram-Schmidt process. The goal is to demonstrate the orthogonality of the vectors \(\vec{v}_Q\) and \(\vec{v}_T\), ensuring they are perpendicular when \(Q \neq T\) within the specified range.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20fbb71c-9fd5-4b59-9900-a3864652c523%2F6b1d7efc-3b3e-43fe-87c2-19229f2e3234%2Fic3jaj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Orthogonal Vector Set Calculation**
**Problem Statement:**
Consider a set of linearly independent vectors \(\{\vec{x}_1, \vec{x}_2, \ldots, \vec{x}_i\}\) in \(\mathbb{R}^n\). Let a vector \(\vec{v}_1 = \vec{x}_1\) and a vector \(\vec{v}_p\) be computed as follows:
**Equation A:**
\[
\vec{v}_p = \vec{x}_p - \sum_{k=1}^{p-1} \left( \frac{\vec{x}_p \cdot \vec{v}_k}{\vec{v}_k \cdot \vec{v}_k} \right) \vec{v}_k, \quad \text{for} \quad 2 \leq p \leq i.
\]
**Objective:**
Show that \(\vec{v}_Q \cdot \vec{v}_T = 0\) for any \(1 \leq Q, T \leq i\) and \(Q \neq T\).
**Explanation:**
This problem involves creating an orthogonal set of vectors from a linearly independent set using the Gram-Schmidt process. The goal is to demonstrate the orthogonality of the vectors \(\vec{v}_Q\) and \(\vec{v}_T\), ensuring they are perpendicular when \(Q \neq T\) within the specified range.
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 { in are linearly independent.
Define:
Observe that:
Step by step
Solved in 2 steps
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