State why (u, v) is not an inner product for u = (ui, u:) and v = (v, V2) in R? %3D (u, v) = (u,v;)² + (u2v2)²
State why (u, v) is not an inner product for u = (ui, u:) and v = (v, V2) in R? %3D (u, v) = (u,v;)² + (u2v2)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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![**Answer the following:**
3. State why ⟨u, v⟩ is not an inner product for \( u = (u_1, u_2) \) and \( v = (v_1, v_2) \) in \( \mathbb{R}^2 \).
\[ \langle u, v \rangle = (u_1 v_1)^2 + (u_2 v_2)^2 \]
---
### Explanation
This problem asks to verify why the given operation is not an inner product in \( \mathbb{R}^2 \). An inner product must satisfy four properties:
1. **Conjugate Symmetry:** \( \langle u, v \rangle = \overline{\langle v, u \rangle} \)
2. **Linearity in the First Argument:** \( \langle au + bw, v \rangle = a\langle u, v \rangle + b\langle w, v \rangle \) for all scalars \( a, b \) and vectors \( u, w \)
3. **Positive Definiteness:** \( \langle u, u \rangle \geq 0 \) with equality if and only if \( u = 0 \)
4. **Additivity in the First Argument:** \( \langle u+w, v \rangle = \langle u, v \rangle + \langle w, v \rangle \)
The given operation \( \langle u, v \rangle = (u_1 v_1)^2 + (u_2 v_2)^2 \) does not satisfy all of these conditions, most likely the positive definiteness or linearity, making it not a valid inner product.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4830dd89-8f3d-4b81-a0b2-1968a7068beb%2F76a73810-b8ff-46c1-81f6-2cfb7114319e%2F8rzsium_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Answer the following:**
3. State why ⟨u, v⟩ is not an inner product for \( u = (u_1, u_2) \) and \( v = (v_1, v_2) \) in \( \mathbb{R}^2 \).
\[ \langle u, v \rangle = (u_1 v_1)^2 + (u_2 v_2)^2 \]
---
### Explanation
This problem asks to verify why the given operation is not an inner product in \( \mathbb{R}^2 \). An inner product must satisfy four properties:
1. **Conjugate Symmetry:** \( \langle u, v \rangle = \overline{\langle v, u \rangle} \)
2. **Linearity in the First Argument:** \( \langle au + bw, v \rangle = a\langle u, v \rangle + b\langle w, v \rangle \) for all scalars \( a, b \) and vectors \( u, w \)
3. **Positive Definiteness:** \( \langle u, u \rangle \geq 0 \) with equality if and only if \( u = 0 \)
4. **Additivity in the First Argument:** \( \langle u+w, v \rangle = \langle u, v \rangle + \langle w, v \rangle \)
The given operation \( \langle u, v \rangle = (u_1 v_1)^2 + (u_2 v_2)^2 \) does not satisfy all of these conditions, most likely the positive definiteness or linearity, making it not a valid inner product.
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