Re: Questions, Week 8 Notes

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Posted by Terence Tao on December 01, 2002 at 20:03:43:

In Reply to: Questions, Week 8 Notes posted by Michael on November 29, 2002 at 23:31:55:

>On page 23, should the result be ||cv|| = |c| ||v|| instead of |c| |v|? If v is a vector, does |v| have any meaning, or just ||v||?

Oops! Sorry about that, |v| should be ||v||. (Though if
one were to wrote |v|, people would probably be able to
understand what you meant, though they may then think that
v should be a scalar or something).

>On page 24, can we say -c ||v||^2 = -||v|| ||-cv|| only because we know ||-cv|| = ||cv||, and thus -||v|| ||-cv|| = -||v|| ||cv|| = -||v|| c ||v|| = -c ||v||^2?

Yup. Alternatively, one can write c || v || = |-c| ||v|| =
|| -c v ||.

>On page 25, you say |a|^2 ||v||^2 + ab' + ba' ' + |b|^2 ||w||^2 is positive for any choice of scalars a, b. I understand how the first and last terms, because they are squared, are non-negative, and since the sum is greater than 0 then we must have |a|^2 ||v||^2 + |b|^2 ||w||^2 > ab' + ba' '. But i don't understand why this is true.

This is because the quantity you mention is equal to
= || av + bw ||^2, which is always
non-negative.

Terry

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>>On page 23, should the result be ||cv|| = |c| ||v|| instead of |c| |v|? If v is a vector, does |v| have any meaning, or just ||v||? >Oops! Sorry about that, |v| should be ||v||. (Though if >one were to wrote |v|, people would probably be able to >understand what you meant, though they may then think that >v should be a scalar or something). >>On page 24, can we say -c ||v||^2 = -||v|| ||-cv|| only because we know ||-cv|| = ||cv||, and thus -||v|| ||-cv|| = -||v|| ||cv|| = -||v|| c ||v|| = -c ||v||^2? >Yup. Alternatively, one can write c || v || = |-c| ||v|| = >|| -c v ||. >>On page 25, you say |a|^2 ||v||^2 + ab' <v,w> + ba' <v,w>' + |b|^2 ||w||^2 is positive for any choice of scalars a, b. I understand how the first and last terms, because they are squared, are non-negative, and since the sum is greater than 0 then we must have |a|^2 ||v||^2 + |b|^2 ||w||^2 > ab' <v,w> + ba' <v,w>'. But i don't understand why this is true. >This is because the quantity you mention is equal to ><av+bw, av+bw> = || av + bw ||^2, which is always >non-negative. >Terry

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