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Use the Gram-Schmidt process to construct a set of orthonormal vectors from + -⠀ X₁ = X₂ = [. 1-2) [ 1 + ² j X3 =

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

Use the Gram-Schmidt process to construct a set of
orthonormal vectors from
[.
1-2)
[1 + ² j
X₁ =
+
X₂ =
L
X3 =

Expert Solution

Step 1: Complex inner product space

Here the vectors are in $C^{3}$ .

Let $u = (u_{1}, u_{2}, u_{3}), v = (v_{1}, v_{2}, v_{3}) \in C^{3}$

Then the inner product

$⟨ u, v ⟩ = u_{1} \bar{v_{1}} + u_{2} \bar{v_{2}} + u_{3} \bar{v_{3}}$

Where $\bar{z}$ denote the complex conjugate of $z$ .

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