Definition 1 (Inner Product Space). Let V be a vector space over a subfield F of C. An inner product (·,·): V × V → F is a function with the following properties Positive Definite (x,x) ≥0 and (x, x) = 0 if and only if x = 0 Conjugate Symmetry (x, y) = (y,x) (ax, y) = a (x, y) (x + y, z) = (x,z) + (y, z) Linearity ● for all x, y, z EV and a € F. V together with an inner product is called an inner product space. Determine whether the given operation on R² is an inner product where x = [x₁ X2] and y = [y₁ 92] (a) (x, y) = x₁x2 — Y2Y1 (b) (x, y) = 2x1Y1 +8Y2x2
Definition 1 (Inner Product Space). Let V be a vector space over a subfield F of C. An inner product (·,·): V × V → F is a function with the following properties Positive Definite (x,x) ≥0 and (x, x) = 0 if and only if x = 0 Conjugate Symmetry (x, y) = (y,x) (ax, y) = a (x, y) (x + y, z) = (x,z) + (y, z) Linearity ● for all x, y, z EV and a € F. V together with an inner product is called an inner product space. Determine whether the given operation on R² is an inner product where x = [x₁ X2] and y = [y₁ 92] (a) (x, y) = x₁x2 — Y2Y1 (b) (x, y) = 2x1Y1 +8Y2x2
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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![Definition 1 (Inner Product Space). Let V be a vector space over a subfield F of C.
An inner product (·,·): V × V → F is a function with the following properties
Positive Definite (x,x) ≥0 and (x,x) = 0 if and only if x = 0
Conjugate Symmetry (x, y) = (y,x)
Linearity
(ax, y) = a (x, y)
●
(x + y, z) = (x,z) + (y, z)
●
for all x, y, z EV and a € F.
V together with an inner product is called an inner product space.
Determine whether the given operation on R2 is an inner product where x = [12]
and y = [y₁ 9₂]
(a) (x, y) = x₁x2 - Y2Y1
(b) (x, y) = 2x1y1 +8Y2x2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2903bb2a-771d-40b7-a62d-703de2e08baa%2F97a8d2e3-d504-405c-9ae8-38ba6d34bd46%2Fqx7rzn9_processed.png&w=3840&q=75)
Transcribed Image Text:Definition 1 (Inner Product Space). Let V be a vector space over a subfield F of C.
An inner product (·,·): V × V → F is a function with the following properties
Positive Definite (x,x) ≥0 and (x,x) = 0 if and only if x = 0
Conjugate Symmetry (x, y) = (y,x)
Linearity
(ax, y) = a (x, y)
●
(x + y, z) = (x,z) + (y, z)
●
for all x, y, z EV and a € F.
V together with an inner product is called an inner product space.
Determine whether the given operation on R2 is an inner product where x = [12]
and y = [y₁ 9₂]
(a) (x, y) = x₁x2 - Y2Y1
(b) (x, y) = 2x1y1 +8Y2x2
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