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Problem 2. Let {x;}"_, be a sequence of vectors in R" with n > m. Assume that there exists a constant A > 0 such that for all x E R". (1) i=1 Define a linear operator S on R" by n Sx = > (x, xi) xi for all æ € R". i=1 Prove the following: 1. span{x1,..., æn} = R".

Problem 2. Let {x;}"_, be a sequence of vectors in R" with n > m. Assume that there exists a constant A > 0 such that for all x E R". (1) i=1 Define a linear operator S on R" by n Sx = > (x, xi) xi for all æ € R". i=1 Prove the following: 1. span{x1,..., æn} = R".

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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Problem 2. Let {x;}_1 be a sequence of vectors in Rm with n > m. Assume that there exists a constant A >0 such that n for all x E R". (1) i=1 Define a linear operator S on R™ by n Sx = (x, x;) Xi for all æ E R". i=1 Prove the following: 1. span{x1,... , xn} = R™. 2. S is self-adjoint. 3. A ||x||? < (Sx, x) for all r E R".
Transcribed Image Text:Problem 2. Let {x;}_1 be a sequence of vectors in Rm with n > m. Assume that there exists a constant A >0 such that n for all x E R". (1) i=1 Define a linear operator S on R™ by n Sx = (x, x;) Xi for all æ E R". i=1 Prove the following: 1. span{x1,... , xn} = R™. 2. S is self-adjoint. 3. A ||x||? < (Sx, x) for all r E R".
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