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2. An inner product on a vector space V is any mapping ,-) V x V R with the following properties: i. Positive definiteness: (x, x) 0 for all x E V and (r, x) 0 if and only if x 0 ii. Symmetry: (x, y) = (y, ) for all r, yE V ii. Linearity: (axby, z) a(x, z)b(y, z) for all r,y, z E V and a, b e R. Show the following a) The dot product y= iyi is an inner product on the vector space R". i-1 To (b) The product (x, y) = x(t)y(t) dt is an inner product on the vector space of functions

2. An inner product on a vector space V is any mapping ,-) V x V R with the following properties: i. Positive definiteness: (x, x) 0 for all x E V and (r, x) 0 if and only if x 0 ii. Symmetry: (x, y) = (y, ) for all r, yE V ii. Linearity: (axby, z) a(x, z)b(y, z) for all r,y, z E V and a, b e R. Show the following a) The dot product y= iyi is an inner product on the vector space R". i-1 To (b) The product (x, y) = x(t)y(t) dt is an inner product on the vector space of functions

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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2. An inner product on a vector space V is any mapping ,-) V x V R with the following properties: i. Positive definiteness: (x, x) 0 for all x E V and (r, x) 0 if and only if x 0 ii. Symmetry: (x, y) = (y, ) for all r, yE V ii. Linearity: (axby, z) a(x, z)b(y, z) for all r,y, z E V and a, b e R. Show the following a) The dot product y= iyi is an inner product on the vector space R". i-1 To (b) The product (x, y) = x(t)y(t) dt is an inner product on the vector space of functions
Transcribed Image Text:2. An inner product on a vector space V is any mapping ,-) V x V R with the following properties: i. Positive definiteness: (x, x) 0 for all x E V and (r, x) 0 if and only if x 0 ii. Symmetry: (x, y) = (y, ) for all r, yE V ii. Linearity: (axby, z) a(x, z)b(y, z) for all r,y, z E V and a, b e R. Show the following a) The dot product y= iyi is an inner product on the vector space R". i-1 To (b) The product (x, y) = x(t)y(t) dt is an inner product on the vector space of functions
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