Exercise 3.2.4. Show that the Schwarz inequality 
is an equality if and only if 
and 
are on the same line through the origin. Describe the situation if and are on the opposite sides of the origin.
Answer: We assume that both and are nonzero. (If either 
or 
then 
and either 
or 
so that 
. So in this case it is trivially true that 
.) We first show that if then and are on the same line through the origin.
If then we have 
. (We can do the division because per our assumption above both and are nonzero and thus the product of their lengths is nonzero.) The denominator is always positive, but the numerator can be either positive or negative. If it is positive then we have 
and if it is negative then we have 
. But 
where 
is the angle between and , so we have either 
or 
.
In the former case the angle 
(or more generally, 
for some integer 
), and and lie on the same line through the origin, on the same side of the origin. In the latter case the angle 
(or more generally, 
for some integer ), and and lie on the same line through the origin, but on the opposite side of the origin.
We next show that if and are on the same line through the origin then .
If and lie on the same line through the origin, on the same side of the origin, then the angle between and is 0. We then have 
so that 
. If and lie on the same line through the origin, on the opposite side of the origin, then the angle between and is 
. We then have 
so that 
. Combining the two equations we have .
We have thus shown that if and only if and are on the same line through the origin.
As implied by the equations above, if and lie on the same line through the origin, on the opposite side of the origin, then 
and the value 
is negative.
NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition
by Gilbert Strang.
If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition
, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition
and the accompanying free online course, and Dr Strang’s other books
.
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