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 vEV, 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=lv-proja(v)l then u = proja(v)

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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)u1 + (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,

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)u1 + (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,

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 41CR: Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the...

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