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Let V = {(x, y) x, y = R}. Suppose the vector addition and scalar multiplication in V are defined the same way as in standard R2. Let u = (u₁, u₂), v = (v₁, v2) be vectors in V. We define an inner product in V according to the formula (u, v) = ₁₁-₁2-21 +222. Show that V equipped with this inner product is a real inner product space.

Let V = {(x, y) x, y = R}. Suppose the vector addition and scalar multiplication in V are defined the same way as in standard R2. Let u = (u₁, u₂), v = (v₁, v2) be vectors in V. We define an inner product in V according to the formula (u, v) = ₁₁-₁2-21 +222. Show that V equipped with this inner product is a real inner product space.

Advanced Engineering Mathematics
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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Let V = {(x, y) x, y = R}. Suppose the vector addition and scalar multiplication in V are defined the same way as in standard R². Let u = (₁, ₂), v = (v₁, V₂) be vectors in V. We define an inner product in V according to the formula (u, v) = 1 V2 — U2V₁ + 2U2V2. Show that V equipped with this inner product is a real inner product space.
Transcribed Image Text:Let V = {(x, y) x, y = R}. Suppose the vector addition and scalar multiplication in V are defined the same way as in standard R². Let u = (₁, ₂), v = (v₁, V₂) be vectors in V. We define an inner product in V according to the formula (u, v) = 1 V2 — U2V₁ + 2U2V2. Show that V equipped with this inner product is a real inner product space.
Expert Solution
Step 1: Given Data

Let V equals left curly bracket left parenthesis x comma y right parenthesis vertical line space x comma y element of straight real numbers right curly bracket. 

Let, u equals left parenthesis u subscript 1 comma u subscript 2 right parenthesis comma space v equals left parenthesis v subscript 1 comma v subscript 2 right parenthesis be vectors in V. We define an inner product in V according to the formula:

less than bold italic u bold comma bold italic v bold greater than equals u subscript 1 v subscript 1 minus u subscript 1 v subscript 2 minus u subscript 2 v subscript 1 plus 2 u subscript 2 v subscript 2

We have to show that V equipped with this inner product is a real inner product space.

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