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
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
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 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)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93785f8e-4f6f-4e45-a8fa-13693ab25e29%2F4b37a724-41b8-49c9-89ee-29d7fc5add50%2Ff18epi_processed.png&w=3840&q=75)
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 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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