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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,

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,(*,*)) 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 vEV, 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=lv-proja(v)l then u = proja(v)
Transcribed Image Text: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 vEV, 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=lv-proja(v)l then u = proja(v)
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