Let (V,(*,*)) be an F-inner product space,where F is either R or C. LetU C V be a subspace andlet a .= {u1,u2,...,uk} be an orthonormal basis for U. For each v E V, we definedproja(v) .= (v,u1)ui + (v,u2)u2+ ·… + (v,uk)uk.Prove that proja(v) is the closest vector to v in U and that it is the unique such vector, i.e. for all uEU,a) Iv – ul Iv – proja(v)lb) if Iv-ul=Iv-proja(v)l then u = proja(v)

To show, projαv is the closest vector to v in U and that it is unique such that, v-u≥v-projαv if…

Answered: Sui+ (v,u2)u2+ ··+ (v,uk)uk. =(v) is…

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Sui+ (v,u2)u2+ ··+ (v,uk)uk. =(v) is the closest vector to v in U and that it is the Iv - proja(v)I

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 54CR: Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector...

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Question

Let (V,(*,*)) be an F-inner product space,where F is either R or C. LetU C V be a subspace and
let a .= {u1,u2,...,uk} be an orthonormal basis for U. For each v E V, we defined
proja(v) .= (v,u1)ui + (v,u2)u2+ ·… + (v,uk)uk.
Prove that proja(v) is the closest vector to v in U and that it is the unique such vector, i.e. for all u
EU,
a) Iv – ul Iv – proja(v)l
b) if Iv-ul=Iv-proja(v)l then u = proja(v)

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